I'll be basing all of this on only the Masoretic Hebrew Text, no Septuagint or Hellenistic apocryphal writings. And nothing from Christian translations Jews would object to. I'll be using this linked below Translation made by Jews. Rather then my usual KJV defaulting.
http://www.mechon-mamre.org/p/pt/pt0.htmThe objection to the Incarnation clearly isn't that it's something God can't do. Even Jewish interpretations of Scripture agree that "The Angel of The LORD" or "The Angel of God" is God taking a physical tangible Human looking form. The Word (Dabar in Hebrew) could be viewed that way also, like in Genesis 15 where the Dabar performs the Covenant ritual all on his own. The Angel who announces the conception of Samson is an example, and the Captain of The Host who appears to Joshua before The Battle of Jericho. Anytime an Angel accepts Worship and yet isn't evil or fallen that's clearly not an ordinary Angel but God Himself.
The very name of Israel comes from when Jacob wrestled with God. And in Genesis 18 everyone agrees the leader of the three Angels there is God himself talking with Abraham. And He actually eats food with Abraham and Sarah. That's a pretty physical tangible Human like form, I'd argue incarnating as a Human isn't that much greater a leap.
But again, it's not about what God can do but what he's willing to. Is actually becoming a Human simply too far beneath Him? Remember God made Adam in his own Image, Genesis 1:26. So really why assume incarnating as a Human is something he'd never do when Adam was modeled after himself to begin with?
Then there is the Hebrew word Go'el. That word is variously translated Kinsman, Redeemer, and Avenger/Revenger. The word means all of those things. It maybe does not necessarily literally have to mean a biological relative every time it's used, but the Kinsman aspect is important to it's function in the Mosaic Law. And is vital to understanding the Book of Ruth, where Boaz is the Kinsman Redeemer, being a near male relative of Naomi and Ruth's late husband.
The word is used of God in Isaiah 41:14 and 43:14 "Fear not, thou worm Jacob, and ye men of Israel; I help thee, saith the LORD, and thy Redeemer, the Holy One of Israel." Also in 44:6 and 24. And 47:4, 48:17, 49:7 and 26 and 54:5. And other Isaiah examples, also Jeremiah 50:34
Job said in 19:25 "But as for me, I know that my Redeemer liveth, and that He will witness at the last upon the dust". Also Psalms 19:14 and 78:35.
What about the Preexistence of The Messiah?
Micah 5 "out of thee shall one come forth unto Me that is to be ruler in Israel; whose goings forth are from of old, from ancient days." and Isaiah 9:5 "For a child is born unto us, a son is given unto us". Both verses even in Jewish translations imply a Preexistence.
Rapheal Patai is a Jewish scholar who agrees to the Preexistence of The Messiah "The concept of the preexistence of The Messiah accords with the general Talmudic view which holds that "The Holy One, blessed be He, prepares the remedy before the wound"", (The Messiah Texts pp. 16-17). Preexistence alone doesn't prove Divinity, but it makes him very special. Because while some cults believe we all had a preexistence like the Mormons, that view is entirely UnBiblical, from Genesis 2 "Then 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." The Soul and Spirit are created at the same time The Body is.
The Messiah is David's Son/Descendant. Yet David calls him lord in Psalm 110 "The LORD saith unto my lord: 'Sit thou at My right hand, until I make thine enemies thy footstool.'" Psalm 2 is also interesting. The Messiah appears to be relating how "the LORD said unto me: 'Thou art My son, this day have I begotten thee. Ask of Me, and I will give the nations for thine inheritance, and the ends of the earth for thy possession."
Psalm 45 is considered a Messianic Psalm, it explicitly refers to the King as God in verse 8, and says in verse 11 to Worship him.
The clincher however I believe is to look at Ezekiel 40-48's description of the coming Messianic Kingdom. How come this in depth description mentions no Palace where The Messiah Ben-David rules from? A great deal of the point of the Messianic Age is to fulfill the Davidic promise from II Samuel 7, that a Son of David would sit on David's Throne forever. And this promise is inherently linked to Jerusalem.
And yet Ezekiel in his in-depth description of The Messianic future geography and architecture of Israel, Jerusalem and The Temple mentions no dwelling place for The Messiah. The only Throne mentioned is in Ezekiel 43:7 in the Holy of Holies, no longer separated from The Holy Place by the Veil. Where The LORD tells Ezekiel "this is the place of My throne, and the place of the soles of My feet, where I will dwell in the midst of the children of Israel for ever; and the house of Israel shall no more defile My holy name".
Here the LORD is saying he himself will rule. How can this be reconciled with the Davidic Promise? Clearly the Throne of David and the Throne of God have become the same Throne. And therefor God must incarnate as The Son of David, Son of Abraham, Son of Adam.
Ezekiel 40-48 does have references to a "Prince" (Nasi in the Hebrew). If the word for Prince here had been Sar or Nagyid then it could make sense to say he's The Messiah, but Nasi isn't a royal term, and could more accurately be translated President. Ezekiel 34:23&24 and 37:24:25 explain that the Nasi is David himself Resurrected, not his Son who's The Messiah.
The LORD also enters through the Eastern Gate, just as The Messiah is supposed to do.
Add on top of that some interesting material from Zechariah 12-14. In 12:17 "In that day shall the LORD defend the inhabitants of Jerusalem; and he that stumbleth among them at that day shall be as David; and the house of David shall be as a godlike being, as the angel of the LORD before them."
That verse is definitely translated differently in Christian translations, but even the way it's translated here is still pretty compelling. Also 14:9 "And the LORD shall be King over all the earth; in that day shall the LORD be One, and His name one." And in verse 16 "And it shall come to pass, that every one that is left of all the nations that came against Jerusalem shall go up from year to year to worship the King, the LORD of hosts, and to keep the feast of tabernacles."
Genesis 3:15's Seed of the Woman is viewed by Jews as being Humanity, and unlike many Christians I'm not going to object to that. It's relevance here is this crushing the head of the Serpent theme does come up again in the Hebrew Bible. Psalms 74:12-14, 89:10, 91:13, Isaiah 27 and Isaiah 51:9. And in those passages it is The LORD that crushes the heads of serpents.
In Genesis 3:20 Adam calls his Wife's name Eve because she is " the mother of all living". Deuteronomy 5:22(verse 26 in other Bibles) calls YHWH the "Living God". Meaning Eve through one of her many daughter has to become the Mother of God.