Jesus began by telling them, “See to it that no one deceives you. Many will come in my name, claiming, ‘I am He,’ and will deceive many.
This gives me the impression that "my name" means Christ (or Messiah), with divine attributes as per Daniel's "son of man" figure. As 1 John 2:22-23 puts it:
Who is the liar, if it is not the one who denies that Jesus is the Christ? This is the antichrist, who denies the Father and the Son. Whoever denies the Son does not have the Father, but whoever confesses the Son has the Father as well.
So here confessing "the Son" means acknowledging that "Jesus is the the Christ," which seems similar to Php. 2:9-11 to me.
Therefore God exalted him to the highest place and gave him the name above all names, that at the name of Jesus every knee should bow, in heaven and on earth and under the earth, and every tongue confess that Jesus Christ is Lord, to the glory of God the Father.
And since I think Josephus' reference to Jesus in the James passage in Ant. 20.9.1 is genuine, I think it's worth noting that Jesus is said to be "called Christ" there.
... the brother of Jesus, who was called Christ ...
So perhaps "Christ" (with divine attributes) is "the name above all names."