Why does Jesus have to be fully God and fully man? Is it not sufficient to be the son of God and fully man?

Jesus Christ, the Lamb was slain from the foundation of the world (Rev. 13:8). OT saints who believed in God were covered by the efficacy of Christ's death from the foundation of the world. Only but a few had that true faith having not seen Jesus Christ's death on the cross. It was then inevitable for Christ to be revealed to us by His coming to earth (1Peter 1:20). Without the shedding of blood is no remission (Heb. 9:22). And "... it is not possible that the blood of bulls and of goats should take away sins" (Heb. 10:4). "For then must he often have suffered since the foundation of the world: but now once in the end of the world hath he appeared to put away sin by the sacrifice of himself" (Heb. 9:26). The Testament of grace would come into effect only through the death of the testator. "For where a testament is, there must also of necessity be the death of the testator. For a testament is of force after men are dead: otherwise it is of no strength at all while the testator liveth" (Heb. 9:16-17).

Unless Jesus came in Human flesh, there could be no acceptable sacrifice for sin. Unless a sinless human sacrifice was presented to God, God's command that man's death was required in reparation for sin could not be fulfilled:

I am taking the liberty of including excerpts from something I have written on the subject of Salvation it was an effort to explain it in a form which would be interesting, and yet give the requirements for Salvation.

I hope you will find it both informative and enjoyable.

How does Jesus death on the Cross give us Salvation?

You see it's like this in Genesis 2:7 is the answer.

Genesis 2:7 And the LORD God formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul.

that Scripture has three separate parts, first God created man from dust, God breathed into his nostrils the breath of life, and finally the most important part in that man became a living Soul. For now we need to consider that part about breathing the breath of life into his nostrils. God never breathed the breath of life into any other human being, yes he did breathe the breath of life into the animals, but that was not repeated either with successive animals or man.

the breath of life was breathed into man by God.

since God is eternal and the breath of life is a part of God, therefore the breath of life is also eternal. That is why he did not breathe it into Eve, because life already existed in the rib taken from Adam.

That same life is passed on to successive generations from parents to children. God referred to descendants as seed.

Genesis 3:15 And I will put enmity between thee and the woman, and between thy seed and her seed; it shall bruise thy head, and thou shalt bruise his heel.

Genesis 4:25 And Adam knew his wife again; and she bare a son, and called his name Seth: For God, said she, hath appointed me another seed instead of Abel, whom Cain slew.

We have to ask if God is a just God why would he punish us for what Adam and Eve did? The answer to that question, because God is a just God he gives us a chance to avoid that punishment.

After Adam and Eve messed up and disobeyed God in order to provide some way 
to let us escape a sinless sacrifice had to be provided. 

But there was a big problem: now that they had the knowledge of good and evil, any seed that they might have would also have that knowledge and even though just having that knowledge in itself is not the reason man has to die; (it is sin {disobedience to God} that condemns man to die).

God knew that man would not be able to withstand the temptations of Satan, and since all men were destined to sin, How could man ever be reconciled to God?

There is only one way that man can ever be reconciled to God, and that being 
that every man must die that God's word in Genesis 2:7 be fulfilled:

Genesis 9:4 But flesh with the life thereof, which is the blood thereof, shall ye not eat.

God says that the blood is the life, and so only the shedding of blood(the life) must be shed in order for man to be reconciled to God, but not just any blood it must be a certain blood. That must be the blood which contains the life God breathed into man at creation.

Since all men's hearts are inherently evil how could man ever be reconciled to God?

The answer to that is that it must be innocent blood, or if you will a life which never sinned (disobeyed God) God showed us in the book of Leviticus:

Leviticus 4:3 If the priest that is anointed do sin according to the sin of the people; then let him bring for his sin, which he hath sinned, a young bullock without blemish unto the LORD for a sin offering.

Leviticus 4:23 Or if his sin, wherein he hath sinned, come to his knowledge; he shall bring his offering, a kid of the goats, a male without blemish:

Leviticus 4:28 Or if his sin, which he hath sinned, come to his knowledge: then he shall bring his offering, a kid of the goats, a female without blemish, for his sin which he hath sinned.

Leviticus 4:32 And if he bring a lamb for a sin offering, he shall bring it a female without blemish.

Leviticus 5:15 If a soul commit a trespass, and sin through ignorance, in the holy things of the LORD; then he shall bring for his trespass unto the LORD a ram without blemish out of the flocks, with thy estimation by shekels of silver, after the shekel of the sanctuary, for a trespass offering:

Any sacrifice to the LORD must be without blemish.

Jesus is our unblemished sacrifice.

Jesus was a man and Jesus was also Deity,

Jesus the man was tempted in every way that man is tempted by Satan, and having also the life form of deity was able to withstand Satan's temptations.

Jesus the man, in dying on the cross; fulfilled God's demand that man must give back that life which he breathed into the first man (or that he must die for his disobedience).

It allows you to have a pathway to salvation.

It is very important that you realize that Salvation is something you must accept and that you too must die on the Cross along with Jesus.

You give up the life you inherited through your parents and take up the new life which Jesus gives you.

John 6:27 Labour not for the meat which perisheth, but for that meat which endureth unto everlasting life, which the Son of man shall give unto you: for him hath God the Father sealed.

John 10:28 And I give unto them eternal life; and they shall never perish, neither shall any man pluck them out of my hand.

God breathed the breath of life into Man (Adam), but God: nowhere else, breathed life into any other human being. That includes Eve who was made from Adam's rib, this is because life already existed in Adam's rib.

God told man (not just Adam alone) that that life would be demanded for his disobedience.

Everyone from Adam to the newest newborn shares that same breath of life God breathed into the first man.

That includes the man Jesus, (but not the Deity Jesus which was alive in the beginning). That was the LIFE that Jesus sacrificed on the Cross! That fulfilled God's demand of life in reparation for SIN.

Salvation comes when we allow our life inherited from Adam to be also sacrificed on the same cross with Jesus.

We do this when we ask Jesus to live within us, as Paul said in;

Galatians 2:20 I am crucified with Christ: nevertheless I live; yet not I, but Christ liveth in me: and the life which I now live in the flesh I live by the faith of the Son of God, who loved me, and gave himself for me.

So why not just keep the eternal life you got through my parents?

So you remember that back at the start I said that Genesis 2:7 had three parts.

Do you also remember that I said the most important part was the part about man becoming a living soul?

The Soul is the part we really need to worry about because that is the part which will continue throughout eternity.

Once a soul is created it cannot ever die, and that is why Jesus gives us a new eternal life, or if you will a new eternal being.

Remember that the breath of life which created the Soul in the first place.

When we accepted Salvation also sacrificed our old life on that cross too.

A soul must have a life associated with it.

God cannot abide Sin, and so no sinners will be in his presence. Any Soul which has sin will have to be somewhere else, and it is that somewhere else that is so horrible we should avoid it at all costs.

John 1:1-3 says that He (the Word) was with God and that he was God. All things were created through Him and without Him nothing was made that had been made. He can't make Himself. On top of that nothing and no one can turn aside God's wrath but God. Therefore he had to be fully God.

I very much like @JamesT's answer, and also disagree with it. Here is where I agree:

God is free. He was under no compulsion to create us, and after we sinned, was under no compulsion to save us. In our desire to worship God's omnipotence and wisdom, it is fitting to ascribe to him the capability of identifying a multitude of ways by which his power and love might accomplish the task. However, he has communicated to us the necessity of some things that he has done. Here are a few ways that God has informed us about necessary things:

Isaiah 63:

4 It was for me the day of vengeance; the year for me to redeem had come. 5 I looked, but there was no one to help, I was appalled that no one gave support; so my own arm achieved salvation for me, and my own wrath sustained me. 6 I trampled the nations in my anger; in my wrath I made them drunk and poured their blood on the ground.”

If God desires that salvation be accomplished, he must do it. The savior must be God.

In Luke 22:

41 He withdrew about a stone’s throw beyond them, knelt down and prayed, 42 “Father, if you are willing, take this cup from me; yet not my will, but yours be done.” 43 An angel from heaven appeared to him and strengthened him. 44 And being in anguish, he prayed more earnestly, and his sweat was like drops of blood falling to the ground.

The cup that Jesus begs to avoid is his death. What father, loving his Son infinitely, on hearing a tearful plea from a Son who has honored him with perfect obedience and love, would reject such a plea if it were optional? The necessity is here made emotional, visceral, and wholly without doubt.

In Genesis 22, another father who passionately loved his son met a different fate:

6 Abraham took the wood for the burnt offering and placed it on his son Isaac, and he himself carried the fire and the knife. As the two of them went on together, 7 Isaac spoke up and said to his father Abraham, “Father?”

“Yes, my son?” Abraham replied.

“The fire and wood are here,” Isaac said, “but where is the lamb for the burnt offering?”

8 Abraham answered, “God himself will provide the lamb for the burnt offering, my son.” And the two of them went on together.

9 When they reached the place God had told him about, Abraham built an altar there and arranged the wood on it. He bound his son Isaac and laid him on the altar, on top of the wood. 10 Then he reached out his hand and took the knife to slay his son. 11 But the angel of the Lord called out to him from heaven, “Abraham! Abraham!”

“Here I am,” he replied.

12 “Do not lay a hand on the boy,” he said. “Do not do anything to him. Now I know that you fear God, because you have not withheld from me your son, your only son.”

13 Abraham looked up and there in a thicket he saw a ram[a] caught by its horns. He went over and took the ram and sacrificed it as a burnt offering instead of his son. 14 So Abraham called that place The Lord Will Provide. And to this day it is said, “On the mountain of the Lord it will be provided.”

We are shown two times: one when an alternate sacrifice was available, and one where an alternate sacrifice was not. The sacrifice of Jesus was not optional.

When Jesus met the two men on the road to Emmaus in Luke 24:

25 He said to them, “How foolish you are, and how slow to believe all that the prophets have spoken! 26 Did not the Messiah have to suffer these things and then enter his glory?” 27 And beginning with Moses and all the Prophets, he explained to them what was said in all the Scriptures concerning himself.

Jesus himself said that the Messiah had to die, and that the whole Old Testament said the same thing. In John, 15:13, Jesus said:

13 Greater love hath no man than this, that a man lay down his life for his friends.

Could Jesus choose not to be our friends? Could he choose not to love to the uttermost degree? If God decided to befriend even one human, and decided to love even one man or woman, being one who shows infinite love, if that friend sinned and needed a savior, then Jesus had to die.

So did Jesus, the Son, have that choice? IN John 10, Jesus says:

11 “I am the good shepherd. The good shepherd lays down his life for the sheep. 12 The hired hand is not the shepherd and does not own the sheep. So when he sees the wolf coming, he abandons the sheep and runs away. Then the wolf attacks the flock and scatters it. 13 The man runs away because he is a hired hand and cares nothing for the sheep.

14 “I am the good shepherd; I know my sheep and my sheep know me— 15 just as the Father knows me and I know the Father—and I lay down my life for the sheep. 16 I have other sheep that are not of this sheep pen. I must bring them also. They too will listen to my voice, and there shall be one flock and one shepherd. 17 The reason my Father loves me is that I lay down my life—only to take it up again. 18 No one takes it from me, but I lay it down of my own accord. I have authority to lay it down and authority to take it up again. This command I received from my Father.”

The father commanded Jesus to take care of the sheep that he had given to him. In John 17, he speaks more plainly:

6 “I have revealed you[a] to those whom you gave me out of the world. They were yours; you gave them to me and they have obeyed your word. 7 Now they know that everything you have given me comes from you. 8 For I gave them the words you gave me and they accepted them. They knew with certainty that I came from you, and they believed that you sent me. 9 I pray for them. I am not praying for the world, but for those you have given me, for they are yours. 10 All I have is yours, and all you have is mine. And glory has come to me through them. 11 I will remain in the world no longer, but they are still in the world, and I am coming to you. Holy Father, protect them by the power of your name, the name you gave me, so that they may be one as we are one. 12 While I was with them, I protected them and kept them safe by that name you gave me. None has been lost except the one doomed to destruction so that Scripture would be fulfilled.

So God the Father gave some group of sheep - people - to Jesus to save and commanded him not to lose any.

From this point, we veer into other questions:

1. What is the nature of the atonement?

2. What is the nature of free will?

3. How great and what kind of debt is owed to God?

Some say that our debt is large but finite and Christ's sacrifice infinite. Thus he went beyond what was necessary, in order to show his love. From that excess merit, Christ has gifts to give us. Some argue that true love requires free will, so Jesus had to be free to not sacrifice his life. Some say that the atonement was not a penal substitution (he did not have to suffer an infinite penalty to take the place of infinitely guilty people). Christ merely was victorious over the devil, sin, death, and they are all finite problems.

Others say that by robbing God of his glory through our disobedience, we are guilty of an infinite offense. This begs the question: would a finite disobedience on our part only require finite love and forgiveness on God's part? In our stinginess, we are programmed to seek to pay only as much as we need to in order to buy a treasure. However, remember the parables of Matthew 13:

44 “The kingdom of heaven is like treasure hidden in a field. When a man found it, he hid it again, and then in his joy went and sold all he had and bought that field.

45 “Again, the kingdom of heaven is like a merchant looking for fine pearls. 46 When he found one of great value, he went away and sold everything he had and bought it.

God is not stingy, nor is his Son. He will sell everything to get what he desires, and he desires to save us. Even if the Father knew it were possible to save us with a lesser sacrifice than His divine Son, he would not. If some other person were to save mankind, they would be worthy of great honor.

In Revelation 5, we hear this:

5 Then I saw in the right hand of him who sat on the throne a scroll with writing on both sides and sealed with seven seals. 2 And I saw a mighty angel proclaiming in a loud voice, “Who is worthy to break the seals and open the scroll?” 3 But no one in heaven or on earth or under the earth could open the scroll or even look inside it. 4 I wept and wept because no one was found who was worthy to open the scroll or look inside. 5 Then one of the elders said to me, “Do not weep! See, the Lion of the tribe of Judah, the Root of David, has triumphed. He is able to open the scroll and its seven seals.”

Here we are given the solemn judgment that NO ONE ELSE IN ALL OF HISTORY WAS WORTHY, except JESUS. To save mankind is an honor that the father gave to his Son to honor him. That honor belongs to no one else.

If I were to continue being true to form, I would answer your question in too many paragraphs, most of which would not be read. Let me go against type, then, and offer a two-paragraph answer to your question,

"Why does Jesus have to be God?"

The reason Jesus has to be God stems from God's holiness. The sin which first entered the celestial universe through Satan, and then through him infected the terrestrial universe via our first parents, was an affront to God, who is infinitely holy. Sin, in turn, triggered within God a holy wrath against sin, and that wrath demanded satisfaction. Whether each person of all the people born since Adam committed one sin or a million sins, the effect would be the same: an infinite chasm between a holy and righteously angry God and unholy sinners (see Psalm 7:11 KJV).

From eternity past, however, God the Father and God the Son had a plan to bridge that infinite chasm between God and man (see Ephesians 1:11 and 3:11), and that was for Jesus, God's infinitely holy Son, to bear at the cross the brunt of God's wrath against sin. Jesus lived a sinless life in perfect obedience to His Father, and at the culmination of that life offered himself up to the Father as the perfect and sinless sacrificial Lamb of God to bear away the sin of the world (John 1:29). Anyone less than God could neither atone for our sin nor through his atonement bridge the infinite gap between us and God. To think that anyone less than God could accomplish this is to underestimate two things: first, the absolute and infinite holiness of the triune God; and second, the utter depravity of humankind. As the apostle Paul realized, there was a need for a double imputation: our sins needed to be imputed to the God-Man, Jesus Christ, and his righteousness needed to be imputed to us, sinners saved by grace through faith (Ephesians 2:8-9). As Paul concluded,

"He who knew no sin became sin for us, that in him we might become the righteousness of God" (2 Corinthians 5:21).

Your question is whether God could, or perhaps should, have chosen to save us other than by becoming incarnate himself.

To begin with, the Incarnation is an act of grace. From the teaching we have received in the sacred scriptures, we know that the coming of God into the world, in the person of Jesus Christ, was a free and unmerited gift of God's love.

God so loved the world that he gave his only Son, so that everyone who believes in him may not perish but may have eternal life. (John 3:16)

God proves his love for us in that while we still were sinners Christ died for us. (Romans 5:8)

There are many more passages I could have quoted. But my point here is that God was in no sense obliged to dwell among us. His sacrifice of himself was freely given, as the ultimate act of love.

The Biblical testimony does show that, in the way it did happen, the joining of human and divine natures in the person of Christ was capable of bringing about our reconciliation with God.

If while we were enemies, we were reconciled to God through the death of his Son, much more surely, having been reconciled, will we be saved by his life. But more than that, we even boast in God through our Lord Jesus Christ, through whom we have now received reconciliation. (Romans 5:10-11)

Might this have been possible if Jesus was not fully God - but only a lesser divinity? The letter to the Hebrews, which you cite, says in chapter 1 that the son of God is more than a mere angel; he is "the exact imprint of God's very being" (v3), temporarily made "lower than the angels" (2:9) in order to be the direct link between God and ourselves. He is more than a servant of God, more than an ordinary prophet or priest or king. This is the free gift of God: he did not send the messenger we asked for, but came himself (9:11ff).

It is this teaching that Leo I expressed in his Tome, famously received by the Council of Chalcedon with the acclamation, "This is what we all of us believe. This is the faith of the Apostles." With more than an echo of Philippians 2, he wrote:

The proper character of both natures was maintained and came together in a single person. Lowliness was taken up by majesty, weakness by strength, mortality by eternity. To pay off the debt of our state, invulnerable nature was united to a nature that could suffer; so that in a way that corresponded to the remedies we needed, one and the same mediator between God and humanity the man Christ Jesus, could both on the one hand die and on the other be incapable of death. Thus was true God born in the undiminished and perfect nature of a true man, complete in what is his and complete in what is ours.

The theology of this letter was also influenced by Augustine, who wrote in De Trinitate 13.13:

Those then who say, What, had God no other way by which He might free men from the misery of this mortality, that He should will the only-begotten Son, God co-eternal with Himself, to become man, by putting on a human soul and flesh, and being made mortal to endure death? - these, I say, it is not enough so to refute, as to assert that that mode by which God deigns to free us through the Mediator of God and men, the man Christ Jesus, is good and suitable to the dignity of God; but we must show also, not indeed that no other mode was possible to God, to whose power all things are equally subject, but that there neither was nor need have been any other mode more appropriate for curing our misery. For what was so necessary for the building up of our hope, and for the freeing the minds of mortals cast down by the condition of mortality itself, from despair of immortality, than that it should be demonstrated to us at how great a price God rated us, and how greatly He loved us?

In this way, Augustine asserts God's sovereign capacity to find some other means for our salvation, but also the fact that the Incarnation was the best possible means - the one most expressive of God's love. Thomas Aquinas, citing these and other authorities (Summa 3.1.2), finds additional reasons, including the idea that the union of natures is essential to lift us above the dominion of evil spirits and the slavery of sin. This is only an elaboration of the fundamental Biblical doctrine of Christ's sacrifice being the supreme act of love, because of his full divinity. A lesser sacrifice would be a lesser expression of the love and grace of God.