Greek, actually - supposedly Palaistina, itself supposedly a transliteration from the Hebrew for “Land of the Philistines” ( Pleshet - פלשתינה ).
Where the Romans came in is in formally naming the larger region the province of Syria Palaestina in 135 C.E. ( originally the term appears to have corresponded to smaller sub-region, but the Greeks appear to have used the term more generally ). This is certainly canonical Roman history and any map of Roman provinces from that period will show the region so-named ( until late in Roman history when it was further subdivided ). However whether the Philistine/Palestine etymology holds up is a different question.
Based on an assumption of Latin or Greek?
Well it was Herodotus who is usually referred to as having first used the term, so that should be easily confirmed ( I don’t have an original in Greek handy, myself 
). But whether he did indeed get the word from the Hebrew for Philistine I have no idea. Perhaps Jomo Mojo can assist with something more definitive :).
- Tamerlane