There’s a fairly strong naming tradition in Italian families:
First son is named after the father’s father
Second son is named after the mother’s father
First daughter is named after the father’s mother
Second daughter is named after the mother’s mother
After that, family names are favored, but there’s no fixed rule. In my family, the naming rules were maintained long after Italian stopped being spoken, and even after intermarriage, so you end up with kids named silly-sounding things like Rocco McSweeney.
My great-grandparents came over from Italy in 1910, but my mother married a non-Italian. However, she still stuck to the naming convention, so I was named after my father’s father (even though it’s not an Italian name) and my brother is named after my mother’s father (which is an Italian name). These rules are why Italians tend to have nicknames - if you follow the rules, then Grandpa Tony ends up with six grandsons named Tony. You have to distinguish them somehow, so they get nicknames - “Fat Tony”, “Short Tony” and such.