“He that shall overcome, i will make him a pillar in the temple of my God; and he shall go out no more; and I will write upon him the name of my God, and the name of the city of my God, the new Jerusalem, which cometh down out of heaven from my God, and my new name” Apocalypse 3:12
This passage mentions the Lord will write his new name on the saints. What is his new name, what was his old name???

The phrase itself is a reference to Isaiah 62.
For Zion’s sake I will not keep silent,
and for Jerusalem’s sake I will not rest,
until her vindication goes forth as brightness,
and her salvation as a burning torch.
The nations shall see your vindication,
and all the kings your glory;
and you shall be called by a new name
which the mouth of the LORD will give.
You shall be a crown of beauty in the hand of the LORD,
and a royal diadem in the hand of your God.
You shall no more be termed Forsaken (Azubah),
and your land shall no more be termed Desolate (Shemamah);
but you shall be called My delight is in her (Hephzibah),
and your land Married (Beulah);
for the LORD delights in you,
and your land shall be married.
‘My new name’ is likely parallel with ‘new Jerusalem’; in other words, Jesus is being equated with the true Jerusalem or Israel.
We aren’t exactly told what the ‘new name’ is at this point. Some think that it is actually referring to the Name
Yhwh - in other words, Jesus is identified with God the Father (“I will write on him the name of my God … and my own new name”). Others meanwhile would point to the later vision in chapter 19.
Then I saw heaven opened, and behold, a white horse! He who sat upon it is called Faithful and True, and in righteousness he judges and makes war. His eyes are like a flame of fire, and on his head are many diadems;
and he has a name inscribed which no one knows but himself. He is clad in a robe dipped in blood,
and the name by which he is called is The Word of God. …]
On his robe and on his thigh he has a name inscribed, King of kings and Lord of lords.
Traditionally, some commentators interpreted the “new name” in a symbolic way: the Name is spoken of as ‘new’ in the sense that in God, there is no such thing as ‘old’. Or, it is ‘new’ in regard to Jesus as the Son of man (His human nature?) The 5th century writer Tyconius (he was a Donatist strictly speaking, but his biblical exegesis influenced orthodox writers like St. Augustine) writes that
“And I will write on him the name of my God.” That is, we are signed with the name of Christian.
“And the name of the city of the new Jerusalem which comes down from my God out of heaven.” The name of the church that daily comes down from God out of heaven, that is, from the church that is reborn by the Lord. He speaks of it as new because of the newness of the Son of man who is the new Jerusalem. … “Indeed, the name which is above every name.” This name is not new because it refers to the Son of God, who existed before the world and possessed this glory with the Father. Rather, this name is new in regard to the Son of man who died and on the third day rose again and, ascending to the heavens, sits at the right hand of the Father.
St. Bede puts it more concisely: “This is the Christian name, not that it is new to the Son of God, who had this glory before the world was made, but new to the Son of Man, who died and rose again, and sitteth on the right hand of God.”