This claim is tied to a missionary favorite, T'hillim / Psalm 110. The controversy begins with the mistranslated “The Lord says to my lord.” Missionaries somehow jump to the conclusion that G-d is speaking to "himself" as Jesus sitting next to him.
Quite a leap in imagination!
This claimed prophecy is refuted by the simple fact that nowhere does it speak of a "son of David" -- which is the claimed prophecy!
Let's begin by reading that first line: "Of David a psalm. The word of the L-rd (G-d's holiest name) to my master (a human in power); "Wait for My right hand, until I make your enemies a footstool at your feet" T'hillim / Psalm 110:1.
Where does this speak of the messiah?
Where does this speak of a son of David?
Matthew 22:43-44 misuses this psalm -- ignoring the context of the psalm, but even so Matthew seems to use this not as a clear statement of anything but rather to ask the question of how King David could call Jesus "lord" if Jesus was his "son" (descendant). Good question, and Matthew does not answer it.
As we take each claim around T'hillim / Psalm 110 individually it will be clear that it is not a messianic prophecy, is not about Jesus and has been badly mangled by translators over the years. For the sake of this claim (son of David) suffice it to say that the psalm says nothing about a son of David, so the claim is false.