If anything?
Care to give an example just for the hell of it?
Perhaps you’re thinking of -ucci, as in Monica Bellucci? That’s much more common. The only Italian surname I can think of off the top of my head that ends in -icci is “Ricci.”
Anyway, -ucci means “descendant of.”
Medici?
It is a diminuitive for example carne is meat carniccio is a scrap of meat. Carniccio is also a surname but presumably from the whole word.
Actress Andrea Marcovicci.
If I can hijack this thread for a bit, I’m now curious if the “Gian” and “Giam” prefixes are comparable. I know Gian is short for Giovanni. Giovammi, maybe? Giampaolo? 
-icci is the old genitive (possessive) of -iccio, which is usually a diminutive suffix.* There are a whole range of diminutive additives with -cci-: -accio (“shades of physical or moral ugliness”), -iccio (“slightly; partly; a little”, used with adjectives); -occio ("-ish"), -uccio (“poverty, meanness, insignificance, pity, or contempt”). Information from Vincent Luciani, “Augmentatives, Diminutives and Pejoratives in Italian,” Italica 20:1 (March, 1943), 17–29.
Giam- is the form Gian- takes before labial consonants: N+P > M+P.
*I believe the common name Ricci has a different origin, for example.
Probably an Italianized name of Slavic origin: Markovic (upside down circonflex on c)
I was not aware that -ucci means “descendant of” until now, but the -i ending indicates masculine plural (changed from the -o of masculine singular).
For example, “ricco” means “rich,” and “ricchi”* means “riches” (or the adjective “rich” in the plural). There is an Italian restaurant in D.C. called I Ricchi (The Riches, or possibly The Rich [Ones]) which I believe is a bit of pun on the name of the owner, Francesco Ricchi.
An ending such as “icci” indicates that it’s plural of a word that ends in “iccio”. There may be some cases where the ending may add some sort of meaning as Erdosain mentions, but it’s just one of many common word endings in Italian. Names can have a diminutive ending, but so can other words.
Also, note that some Italian spellings may have been corrupted from the original, victims of crossing the Atlantic.
*Italian spelling rules dictate that the “h” be added to maintain the hard “c” sound here.
Not always. In surnames, it’s often a fossilized genitive singular, a case that no longer exists in Italian nouns, equivalent to English words in -’s. It is identical in form to the plural (as with English -s). Think of the surname “Williams” – it doesn’t mean more than one “William,” though you can see why it would look that way.
Didn’t even know that in English, and can’t imagine what it does mean.
What does “fossilized genitive singular” mean?
Others can explain this better than me, and I hope they’ll chime in.
- Child (kid): singular
- Children (kids): plural
- Child’s (kid’s): genitive (possessive) singular
- Children’s (kids’): genitive (possessive) plural
So:
- William
- Williams [more than one]
- William’s [dog, child, whatever]
- Williams’ [dog, child, whatever, belonging to more than one William]
The surname is choice #3, spelled like choice #2.
“Fossilized” means an archaic form that has been fixed, even though we’re no longer using those rules. Like “whom,” for instance. Most people only use “who” in colloquial speech, but they might still say “to whom it may concern,” because they’ve just learned the whole phrase. It used to be in English that we’d add an -m to the dative case (I think, my Old English is rusty). Now we only have the forms “him,” “them,” and sometimes “whom” (and a few others like “whilom”). We don’t add it anymore except in those things that have become words in their own right.