-akis is the patronymic suffix used in Crete; it’s a diminutive, like most patronymics in Greek surnames are.
The surname in Greek is Galifianakis Γαλιφιανάκης or Galyfianakis Γαλυφιανάκης; I see the upsilon surname much more frequently online (except with reference to Zach himself). Galifianos means “from Galifa”; there are references online to a Galifian carnival, ΓΑΛΥΦΙΑΝΟ ΚΑΡΝΑΒΑΛΙ 2012. Galifa in turn is a village in Crete near Iraklio: Γαλίφα Ηρακλείου – Βικιπαίδεια. The village is first mentioned in a census from 1538, and has a current population of 250. The village is spelled with both iota and upsilon.
I don’t know the etymology of the village name, but the related adjective γαλίφης “flatterer” comes from the Italian gaglioffo. An Italian name makes sense as Crete was under Venetian rule; not sure why a village would be called “female flatterer” though.