The church doesn’t have a name. The expression church of Christ is not a name but a description.
It means “the called out ones that belong to the Anointed One.”
It is a description just as churches of the Gentiles is a description (Romans 16:4/ Romans 16:16).
Romans 16:4 (NAS): who for my life risked their own necks, to whom not only do I give thanks, but also all the churches of the Gentiles;
Romans 16:16 (NAS): Greet one another with a holy kiss. All the churches of Christ greet you.
Some cite Acts 4 to justify using a “name” with Jesus’s name in it such as church of Christ:
Acts 4:12 (NAS): And there is salvation in no one else; for there is no other name under heaven that has been given among men by which we must be saved.
There are two problems with this:
The first is that there is no name of the church. The first century Christians didn’t have church signs to bear a name like we do today. This idea is literally denominational (to give a name to).
Secondly, Christ isn’t Jesus’s name; it is a title. It means “the anointed.” Many in the Bible bore this title: priests and kings (both Gentile and Jewish). Jesus, as God’s Son and both king and priest, is the ultimate Anointed One.
In the Bible, many descriptions are used: the way, church of God, the church, etc.
None of these are names.