I always thought it was something like ciccio - ciccione; Vito - Vitone.. sort of like a diminutive or belittling form.
Vitone literally means "big Vito", the opposite of a diminutive. The words in
-ino are diminutive, those in
-one are the opposite. For example, with the word "gatto" (cat):
gatt
ino - little cat, kitten
gatt
one - big cat
But "ciccione" is a different thing: Ciccio is a diminutive of Francesco, but the word "ciccione" means also "fat one, fatty". Although there is also the word "grasso" for "fat".
That's an off-topic about Italian language though.