A godfather was a honorary title from the Christian faith for an individual who sponsors a child through baptism. Traditionally a godparent would also assume some responsibility for ensuring the child received proper religion education. In the more modern and sometimes less religious usage, the godparents might take some responsibility for the child’s moral upbringing and play the part of a good role-model.
Mahirn, a friend of Josh Pike, was the godfather of Pike's son Christopher. (EV comic: "The Flat, Gold Forever")
In 2373, shortly after his birth, Kathryn Janeway agreed to be the godmother of Q's son q. (VOY episode: "The Q and the Grey")
The title was also associated with leaders of the Mafia. (FASA RPG module: The Orions: Book of Common Knowledge)