Rev 3:12

12 He who overcomes, I will make him a pillar in the temple of my God, and he will go out from there no more. I will write on him the name of my God, and the name of the city of my God, the new Jerusalem, which comes down out of heaven from my God, and my own new name. 

My God – is repeated three times in this verse. While the Spirit of Christ and the Father “are one” in the LORD’s Day, the Father is always the one true God.

I will write on Him the name of My God – the name of Jesus, which is also the name of the “the City of My God, the New Jerusalem.” The body of Christ, the Church of God, must bear His name: “God saves.”   That Christ, Jerusalem, and the righteous, bear the name of God is the understanding of rabbinic theology from at least the 1st Century BC. From the Talmud, we read: “Said R. Samuel Bar Nahmani said R. Yohanan, ‘There are three who are called by the name of the Holy one, blessed be he, and these are they: the righteous, the Messiah and Jerusalem.’”1Christ was prophesied to share the name of Jerusalem in the Book of Jeremiah, compare Jer 23:6; and Jer 33:16, where the name is “the Lord is our righteousness.”

My new name – after Christ endured the cross, God gave Him “the name above every name,” (Phil 2:9). This is the glory and reputation that the saints will share if they also overcome; see John 17:22.