Why does Jesus have to be fully God and fully man? Is it not sufficient to be the son of God and fully man? This is a follow on from the question
Why does the Messiah have to be God?

Fully Man: I agree that Jesus had to become man, I believe this is VERY well supported in the bible. Hebrews 2:17-18, Hebrews 4:14-15 and much more.

Fully God? I know that Jesus himself claimed deity, (See all the references here: Moody Global Ministries).

Question: What is the Biblical basis for saying that  Jesus must be fully God and fully man? Why does Jesus have to be God himself?

Is it not sufficient that he is the son of God? The son of God can still save, the son of God can still be our intercessor, the son of God can still have fully God attributes (because he is the only son of God), the son of God can still exist from before there was time. So back to the question, WHY does Jesus have to be God?
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 &quot;the exact imprint of God's very being&quot; (v3), temporarily made &quot;lower than the angels&quot; (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, &quot;This is what we all of us believe. This is the faith of the Apostles.&quot; 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.