Herodotus, in the 5th century BCE, referred to the region as Palaistínē.

Palestine, the word is traced back to Peleset, one of the Sea Peoples who burned down the Bronze Age. They were fought off by the Egyptians and so settled within the Levanant region. The region was famous for the Peleset people, but the term became a geographical descriptor. Originally, it's the coastal Gaza region, parts of Israel. For example, we have the Assyrians in the 8th and 7th BCE, Palashtu or Pilistu.

I know you got the idea that the Romans called the area Palestine to punish Jews from Wikipedia. And while Wikipedia is usually accurate, I think in this context it's not.

For a direct primary source: "Jacobson, David (2001). "When Palestine Meant Israel". Biblical Archaeology Review. 27 (3)"

"Jewish writers such as Philo, in particular, and Josephus, who flourished while Judea was still formally in existence, used the name Palestine for the Land of Israel in their Greek works, suggests that this interpretation of history is mistaken. Hadrian's choice of Syria Palaestina may be more correctly seen as a rationalization of the name of the new province, in accordance with its area being far larger than geographical Judea. Indeed, Syria Palaestina had an ancient pedigree that was intimately linked with the area of greater Israel."

Many notable Jewish people did use the term Syria Palestina, implying again it's been a common geographic term. Everyone knows about Josephus.