"Whatever you ask in my name, this I will do, that the Father may be glorified in the Son. If you ask me anything in my name, I will do it." John 14:13-14 (ESV)

Was the beginning of the use when read literally by the denotation. It is shown to by more so of a hypothetical sense according to:

Lincoln, Andrew (2005). Gospel According to St John: Black's New Testament Commentaries. Bloomsbury Publishing.

Although it seems nigh impossible you find the definitive first time in history because not everything that happened was recorded or documented in historical records.

I don't think we will know exactly who the first person was who prayed in the name of Jesus, or exactly when this happened. However, in Acts 3:6 we see Peter use the name of Jesus in healing:

Then Peter said, "Silver or gold I do not have, but what I do have I give you. In the name of Jesus Christ of Nazareth, walk."

Even before the Acts of the Apostles, we see the seventy who were sent out to preach and heal, return with the following account:

The seventy returned with joy, saying, "Lord, even the demons are subject to us in Your name." (Luke 10:17)

We also see in the scripture quoted by Victor, we pray in the name of Jesus, in faith, to receive:

"Whatever you ask in my name, this I will do, that the Father may be glorified in the Son. If you ask me anything in my name, I will do it." John 14:13-14 (ESV)

Why do we pray in the name of Jesus?

Because of the complete work of the cross of Calvary, and what the name of Christ represents. Salvation, healing, deliverance, breakthrough, victory, a better life, a hope and a future etc.

i hope this helps!

A definitive answer to this question will be elusive, but here are some preliminary findings that give some idea and hopefully inspire more research.

As we'll see, the exact English phrase, "in Jesus' name, amen," appears to be a fairly recent development, as do its variations. But it appears earlier in other languages, notably Latin and German. Thus, we'll follow this outline:

• Phrase in translation (German, Latin)
• Traditional forms in English
• Exact phrase and its variations in English

Phrase in translation

The earliest example of something similar to "in Jesus' name, amen" that I've found comes from the 5th century, in Latin:

In nomine Iesu Christi, Amen. (source)
^{In Jesus Christ's name, amen.}

And in the early 8th century:

In nomine Domini nostri Iesu Christi, amen. (source)
^{In the name of Jesus Christ our Lord, amen.}

Interestingly, the form also turns up much earlier in German than in English. For example, in 1538:

In den Namen Jesu Amen. (source)

And in 1687:

Im Namen Jesu! Amen! (source)

Traditional forms in English

Prayers using the trinitarian formula – "in the name of the Father, and of the Son, and of the Holy Ghost" – can be found early in church history, and it's thus not surprising that early English prayer books used it. More interesting is the frequent ending of prayers with a variation on "Jesus Christ." Over 50 prayers in the 1689 Book of Common Prayer use his name in a phrase immediately before "Amen," such as:

through Jesus Christ our Lord. Amen. (page 167)
through our Lord and Savior Jesus Christ. Amen. (183)
but only the Name of our Lord Jesus Christ. Amen. (190)
for the honor of our Advocate and Mediator, Jesus Christ. Amen. (40)

Yet the phrase in question does not appear. Nor does it appear in the popular non-conformist Method for Prayer (1710) by Matthew Henry. The influence of such works, particularly the BCP, may have been a significant factor in the relatively late appearance of "in Jesus' name, amen" in English.

Phrase in English

Definitively identifying the earliest use of this phrase in English would require the use of more corpora than I have at my disposal. The University of Michigan's Middle English corpus yields no results, so I turn then to Google Books. There, the earliest variation of this phrase in English appears in 1778:

All which we beg for the sake and in the name of Jesus Christ. Amen. (source)

And the exact phrase for the first time only in 1840:

We ask all in Jesus' name. Amen. (source)

Summary

I haven't uncovered written evidence for the use of this phrase in English prior to the 18th century, even though it appears in other European languages much earlier – as early as the 5th century in Latin.

It's certainly possible that this lack of evidence is due to a limited written record (or access to it). But the fact that my sources have German instances of this phrase in the 16th and 17th centuries, but no English ones, suggests that the influence of early English prayer books helped ensure that "in Jesus' name, amen" was not commonly employed. Apparently only after Protestantism split further and revivals spread did the phrase come into common usage and eventually become ubiquitous.