How do you say “godfather” in Greek?

In the vernacular: νονός or νουνός. From a Hellenistic word νόννος ~ νέννος, which could mean dad, uncle, or granddad (Wiktionary says the Hellenistic word in turn is from Latin nonnus).

“The Godfather. One of the greatest best sellers of all time”. (Yes, Best Seller is untranslated. Why do you ask?)

In the language of the church: ἀνάδοχος, “guarantor”.  In Modern Greek the term has been extended to sponsor of a child:

“Sponsor a child in Palestine.”