What is the biblical basis for the belief that the spirit of a person remains conscious after death?

About consciousness, there are 4 phases:

1. The Earth.
2. The Grave.
3. The Resurrection, and the Day of Judgement.
4. The Afterlife.

It is certain consciousness exists in those except the second one, the grave. There are implications of the Afterlife in Torah:

Genesis 25:8 Then Abraham gave up the ghost, and died in a good old age, an old man, and full of years; and was gathered to his people.

Abraham joined the dead, but without details how he joined them, and this keeps repeating with his descendants like Ishmael [25:17], Isaac [35:29], Israel [49:33], and Moses [Deuteronomy 32:50].

Yet, the second group uses a different term:

Exodus 31:14 Ye shall keep the sabbath therefore; for it is holy unto you: every one that defileth it shall surely be put to death: for whosoever doeth any work therein, that soul shall be cut off from among his people.

Breaking the covenant of Abraham holds an identical term as in Genesis 17:14, but no details whether this damnation includes the grave or not.

The Gospels are slightly extensive about the Hereafter, to the point you can sense a kind of evolution, not about the Hereafter itself, it's just how God planned the revelation to flow:

John 14:24 He that loveth me not keepeth not my sayings: and the word which ye hear is not mine, but the Father's which sent me.

Even that the Israelites had a [thin] belief in the Hereafter, it was not detailed in the Torah like how Jesus speaks about it in front of them as in Matthew 25:41-46, and even when he mentions the covenant:

Matthew 26

28 For this is my blood of the new testament, which is shed for many for the remission of sins.

29 But I say unto you, I will not drink henceforth of this fruit of the vine, until that day when I drink it new with you in my Father's kingdom.

&

John 11

23 Jesus saith unto her, Thy brother shall rise again.

24 Martha saith unto him, I know that he shall rise again in the resurrection at the last day.

Although Lazarus didn't mention what he felt in the grave, did he? ... Not that there's definitely no consciousness in the grave, it's the Gospels that doesn't speak about it.

Being Muslim, I admit I didn't read much in the Epistles, maybe the grave was mentioned in them, but I'm not sure. What I'm referring at; whether the grave was mentioned in the Bible is not an issue, it's noted how the Hereafter was not completely revealed with the Torah, it was gradually revealed after it.

Even Judaism has beliefs about the Hereafter that does not directly stem from the Tanakh, but the oral Torah. I don't think the Bible answers this question, given how Judaism and Christianity are differed about the Afterlife, not the grave.

EDIT:

Just a notable part about it:

John 8

51 Verily, verily, I say unto you, If a man keep my saying, he shall never see death.

52 Then said the Jews unto him, Now we know that thou hast a devil. Abraham is dead, and the prophets; and thou sayest, If a man keep my saying, he shall never taste of death.

53 Art thou greater than our father Abraham, which is dead? and the prophets are dead: whom makest thou thyself?

54 Jesus answered, If I honour myself, my honour is nothing: it is my Father that honoureth me; of whom ye say, that he is your God:

55 Yet ye have not known him; but I know him: and if I should say, I know him not, I shall be a liar like unto you: but I know him, and keep his saying.

56 Your father Abraham rejoiced to see my day: and he saw it, and was glad.

First of all, there is no biblical basis for a spirit in a natural person who is ungodly and unsaved unlike someone saved by the perfect and complete work of Jesus Christ on the cross.

Soul alone or with Spirit?

Unsaved person: body and soul only but no Spirit. Soul is a also known as spirit of life, hence man in Genesis is described as flesh wherein is the spirit of life. So context is king to understand the word spirit whenever it occurs. When we read about Pharaoh's spirit was troubled here is soul, spirit of life in the flesh of Pharaoh (Gen 41:8).

Saved person: body, soul, then dwelling by the Holy Ghost (Holy Spirit) in the new birth (new creation). Make also the distinction between the Holy Ghost dwelling in a person (always saved) and the Holy Ghost coming upon a person (could be saved but not necessarily as Saul of the OT).

Jesus Christ is an exception as He has eternal Spirit (Heb 9:14) referring to His deity. He made his Soul naked for death, he also bare our sins in His own Body.

Keep in mind, in earlier Scriptures, we read about flesh and blood, blood is often linked with the soul there, so flesh and blood means body and soul.

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. (Gen 2:7)

As you can see no Spirit but breath of life (a special word in Hebrew was used here to distinguish man from animals), but not the Spirit here definitely, as the natural man has no Spirit. There is no evidence that man was tripartite at first, he becomes so once born from above by the Spirit.

That which is born of the flesh is flesh; and that which is born of the Spirit is spirit. (John 3:6)

So your question seems to be, given the above, as if directed toward the saved souls with a spirit but not toward unsaved ones who have no spirit but souls. Either way, there is consciousness.

Evidence of consciousness after death

The evidence of that soul being conscious after death is in the parable of the rich man and Lazarus the beggar in Luke 16:

And he cried and said, Father Abraham, have mercy on me, and send Lazarus, that he may dip the tip of his finger in water, and cool my tongue; for I am tormented. But Abraham said, Son, remember that thou in thy lifetime receivedst thy good things, and likewise Lazarus evil things: but now he is comforted, and thou art tormented. (Luke 16:24,25)

Obviously the rich (unsaved) was in pain, had he not been conscious how could he state his torment then, in contrast, Lazarus (saved) was comforted.

This witness was from the NT, we have something about Samuel in the OT while he was in Hades (the part for the righteous not for the evil ones) after his death.

And Samuel said to Saul, Why hast thou disquieted me, to bring me up? And Saul answered, I am sore distressed; for the Philistines make war against me, and God is departed from me, and answereth me no more, neither by prophets, nor by dreams: therefore I have called thee, that thou mayest make known unto me what I shall do. (1Sam 28:15)

Samuel was disquieted while he was in Sheol or Hades (in the earth), this also implicates a conscious change from comfort to discomfort.

Note

Hades part for the righteous ones (who are made righteous by the blood of Jesus Christ not they are righteous themselves), was inside the earth but it is separate from the part of the evil ones, and they were taken by Jesus Christ to paradise above first after His death on the cross.

Another evidence of consciousness and knowledge we see in the Psalms:

Consume [them] in wrath, consume [them], that they [may] not [be]: and let them know that God ruleth in Jacob unto the ends of the earth. Selah. (Ps 59:13)

Here, let them know, indicates there is no thought of annihilation or ceasing to exist.

Well, there are two scriptural evidences I would like to cite as an answer to the OP.

1. Lazarus and the rich man

"22 The time came when the beggar died and the angels carried him to Abraham’s side. The rich man also died and was buried. 23 In Hades, where he was in torment, he looked up and saw Abraham far away, with Lazarus by his side. 24 So he called to him, ‘Father Abraham, have pity on me and send Lazarus to dip the tip of his finger in water and cool my tongue, because I am in agony in this fire.’" (Luke.16:22-24)

The following points are in order with regard to the above incident that was related by the Lord Jesus Christ:

(a) Some believe it's only an inventive parable, not a real historical incident. But there is no evidence in the Scriptures that lends credence to the assumption that Jesus told purely unreal stories to get His point across.

(b) Facts about the existence after death are some of the most important matters to know about. Jesus wouldn't possibly talk about them in such a way that would give a totally wrong impression to His hearers.

(c) As a person who knows history from the time it began Jesus talked about what used to happen to the dead--both good and bad--before His own death and resurrection. Individuals were very much conscious after their death as we can see in the above incident.

2. Paul's dilemma

"21 For to me, to live is Christ, and to die is gain. 22 But if I am to live on in the flesh, this will mean fruitful labor for me; and I do not know which to choose. 23 But I am hard-pressed from both directions, having the desire to depart and be with Christ, for that is very much better; 24 yet to remain on in the flesh is more necessary for your sakes." (Philippians.1:21-24)

Apostle Paul is categorical in stating that there are only two possibilities before him. One, to continue in the body, which is beneficial to the other believers. Two, to die and be with Christ, which is beneficial to himself. He is hard-pressed between those two options! There is no hint of a third option, namely, 'soul sleep' between the two that were before Paul.

If it were true that there is a time period for 'soul sleep' between life and resurrection, following death then why would the apostle Paul avoid talking about that period? Furthermore, how could he wish to leave the body or choose death in order for him to be with his Lord? If death leads into 'soul sleep' until resurrection how can death be considered as gain by Paul? In fact, it would only cut short his productive time in this life by being beneficial to others. If 'soul sleep' were the inevitable step after life then Paul would have chosen to stay alive as long as possible so that he could be of much benefit to the believers, instead of leaving the body and entering into a state of 'soul sleep,' which is of no use to Paul or the other believers he was ministering to.

For Paul death does not mean 'soul sleep.' For him it means 'to be with Christ,' nothing more nothing less! This, of course, is what happens to all those who are made righteous in Christ. This is not just being in a state of 'consciousness' but 'bliss!'

Jesus told the thief on the cross "Today you will be with me in paradise" (Luke 23:43) and Paul said that "to be absent from the body is to be present with the Lord" (2 Cor 5:8).

Also, after Jesus heals Jairus' daughter in Luke 8 it says:

They laughed at him, knowing that she was dead. 54 But he took her by the hand and said, “My child, get up!” 55 Her spirit returned, and at once she stood up. Then Jesus told them to give her something to eat.

Her 'spirit' is here clearly portrayed as separate from her body, though one might argue that this was simply a way of saying that she came back to life.

Biblical instances such as these imply that we will be with Christ immediately after death, though our physical bodies will not yet be resurrected.

The belief that the dead remain conscious is shown in scripture.

This is for the believer, the one who is born-again of incorruptible seed.

For I [Paul] am in a strait betwixt two, having a desire to depart, and to be with Christ; which is far better: Phl 1:23

We are confident, I say, and willing rather to be absent from the body, and to be present with the Lord. 2 Cor 5:8

Where is the Lord? At/on the right hand of God the Father in heaven (Acts 7:55, Heb 8:1)

Paul knows this.

I am the God of Abraham, and the God of Isaac, and the God of Jacob? God is not the God of the dead, but of the living.

And what of the unsaved? This is more controversial, not as to conscious existence, but as to place. In Jewish thought, there is a place called paradise (Abraham's bosom) and Gehenna. This is referenced in the story of Lazarus and the rich man (Luke 16:19-31).

So, we have seen that for all peoples, there is a conscious soul or spirit after death of this earthly body.