Didn't the Romans coin the phrase "Palestine," in the first century BC?
Palestine is probably a transliteration or corruption of Philistine or Philistia, but the Romans are the ones who ground down the various Jewish revolts (the Jews referred to those “Jewish Revolts” as “Roman Wars”), the last of which saw the destruction of the Second Temple (though not the Temple Tax, levied on Jews all over the Empire), and renamed the city of Jerusalem as well as the region. Of course, the Romans had revived the kingdoms as part of their usual strategy of expanding Roman rule in stages. The Herodian dynasty wasn’t Jewish; Cleopatra was a Roman puppet, contrary to anachronistic claims (as well as Octavian’s propaganda), put on the throne and kept there by Roman power — until she, uh, wound up on the losing side in the last stage of the Roman civil war.
Herodotus in the 5th century B.C. uses “Palestine” as a name for part of Syria. Since the name goes back to the name of the Philistines, it’s probably much older than the 5th century. Whether it was used as the official name of the area by any of the empires before the Roman Empire, I don’t know.