I don't know anything about the use of CABIRIA as given name in ancient Rome; maybe others can find famous bearers of that name.
If it is Latin, it could be from "Cabiri", the name of one or several deities of Semitic-Greek origin, meaning "the big/great ones" ("die Großen"). This is what my Latin dictionary has (Menge/Güthling).
A Lexicon of Antiquity (Lexikon der Alten Welt, Düsseldorf 1990) says: Both its cult and name are non-Greek, probably Phrygian. The Greeks were obviously not able to interprete the name. The oldest proof is a play by Aishylos called "Kabeiroi".Hope
this helps a bit.