Early Christians considered it important that God has no name, in contradistinction to pagans who had multiple gods and naturally wanted to know which god the Christians worshiped. Eusebius reports of Attalus, being roasted on an iron chair, that "when asked what was the name of God, he answered, 'God has no name like a human being has'." St. Justin Martyr in his second Apology argues that names are given by one's elders, and hence God has no name. Aristides in his Apology says: "He has no name, for everything which has a name is kindred to things created." After quoting Trismegistus to the same effect, Lactantius writes: "God, therefore, has no name, because He is alone; nor is there any need of a proper name, except in cases where a multitude of persons requires a distinguishing mark, so that you may designate each person by his own mark and appellation. But God, because He is always one, has no peculiar name." (The difference in reasons given suggests that there was a well-established doctrine that God had no name, but the reasons for the doctrine were not universally agreed on.)
The idea of God's namelessness is fruitful. It is true that there is the tetragrammaton", but that seems to have been completely unused by Christians until very recently. The early Christians would have thought that the use of a proper name made God too much like a pagan deity. (And, indeed, there is evidence of pagan deities with names akin to the tetragrammaton, e.g., in Ugaritic texts.) The Jewish cessation of use of a proper name for God, and its systematic oral replacement by "Adonai" or "Elohim", would have been seen not as protection against uttering a name too holy for our sinful lips, but as a deepening of the understanding of monotheism, of God's utter transcendence.
But in a way God has a name. The man Jesus Christ is his name to us. Christ is the Logos, the Word that reveals God, the word pointing towards God (I am reading "pros ton Theon" in John 1:1 in a way complementary to the usual reading of "with God"). But his name is unlike the names of humans. His name is a person, consubstantial with him. Nothing less than himself is sufficient for us to call him by. Yet, like a name, he is made sensible in the incarnation.