Italian names like
share the same suffix “-ni”. What does that mean?
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There is not a single meaning, and -ni cannot be considered to be a single suffix. The three examples you give show at least two different cases.
-ini is often the plural form of an original -ino, which is most of times the suffix for a diminutive form (coltello, knife; coltellino, little knife, cutter).
-oni is, analogously, often the plural of -one, the way augmentative words end (libro, book; librone, big book).
So, some of those surnames might have originally been nicknames for some little or big person (“Antonione” could be a big man whose name is Antonio); and then the plural denoted their relatives.
These are not the only possibilities, though. For instance, -ano (plural -ani) often denotes the place someone comes from (Romano, from Rome; Padovano, from Padua). Several Italian surnames were born like this (Romani, Padovani).
TL;DR: There is not a single -ni suffix; at the very least, one has to consider the last vowel before it (-ani, -ini, -oni).
They share the same suffix, but for different reasons.
Antonioni could be a "gentilizio" (en: gentilitial), that indicates the family's origin (Antonio family).
The others, as DaG said, could indicate nickname or provenance.
If you are curious you can find meaning and origin of Italian surnames here: Cognomix.