Actually the word goes back at least 500 years before the Romans destroyed the Temple...Herodotus refers to Palestinians and Syria Palaestina when describing the Persian Empire. I don’t know if the term was ever in official use before Vespasian, but it derives from the name of the Philistines so must have been coined before they were utterly forgotten. It’s not clear how large an area Herodotus means by the term, perhaps just the area the Philistines had inhabited.
Oh, sure, there are earlier references, but the Phillistines were sea-farers and coast-dwellers (and have no relation to the arabs who call themselves Palestinians, but that is another issue entirely).
Long story short, though, using the name for the entire territory came about because the Romans, in order to basically put us Jews in our place, used the name of our old enemy as the name of the land.
And, since the Romans made the maps, the name has stuck.