John Cooper argues that the term "name" can have four basic denotations: (1) a personal name (e.g. Jehovah or Peter); (2) a proper noun (e.g. Father, God, King); (3) a general designation (e.g. dog, cat, slanderer); (4) any linguistic reference (e.g. "God is light" or "the greatest possible being").
While it seems that the pre-Nicenes generally acknowledged that God had a name in the sense of (3) or (4) above, the early church writers often appear to be denying that God has a name in the sense of (1) or (2) above. I am reminded of Marcus Minucius Felix, who writes in Octavius 18:
"Nor should you seek a name for God: God is His name. We have need of titles in cases where we want to separate individuals from a large group; we use, then the distinguishing mark of personal names. But God is unique; all He has for title is God" (Nec nomen deo quaeras: Deus nomen est. Illic vocabulis opus est, cum per singulos propriis appellationum insignibus multitudo dirimenda est: deo, qui solus est, dei vocabulum totum est).
There is an anonymous God tradition in early Christianity that is manifested in the East and West. The tradition asserted that God does not have a proper name.