I'm not sure there is such a thing here tbh.
There is a list of names that, if you hear, you know the person has been named after a grandparent. (the recipient is rarely happy about this!)
All the Francescas I know who have nicknames are: Fra, Francy, Chicca, (Kikka) or Checca (Kekka).
Interestingly, thinking about it, Franco for Francesco is also old fashioned but not quite as much as Franca. My BIL is Franco and he's 50 something. Younger Francescos tend to be Ciccio (Cheecho) or not shortened at all.
DD has an Italian middle name that she hates because it's a "granny" name. (Caterina)