Going back in history...
The British territory?
The largest part became the Kingdom of Jordan.
The Negev, coast line, Jerusalem, and the Galil ultimately got their old name back and became Israel.
The rest can become a part of the Kingdom of Jordan or Egypt or declare itself independent, I don't care; and neither does Israel.
The official name of the territory was "Palestine (Land of Israel)". It can be seen in Hebrew on British money ("pounds") from the territory and before Israel changed the name of that currency to "Sheqel".
The Roman province?
It used to be called Judaea and was renamed "Palestina" by the pagans to remove the land's connection to G-d.
The province stopped existing at some point during the many wars where Christian Romans, Christian Greeks, Zoroastrian and later Muslim Iranians, and Muslim Arabs fought for it. It ended up becoming a part of Turkey, divided into parts of several provinces.
The Philistine city states around Gaza?
Greek invaders (the word "Philistine" or "Palestinian" comes from the Hebrew for "invader") founded and invaded city states in Canaan, where the Israelites and Phoenicians lived. They eventually assimilated a bit and began speaking a Canaanite language, related to Hebrew and Phoenician.
They fought the Israelites a bit and eventually lost, an event reported in the Bible as King David beating Goliath, an enemy much larger but less well-armed than Israel. Israelis still feel proud of King David and beating larger enemies using superior weaponry is still part of Israeli customs.
There was no "Palestine". The land was inhabited by Canaanite tribes when Abraham arrived. Muslims believe that G-d told Abraham's son Yitzaq to settle in Canaan and his other son Yishmael to migrate to Arabia. This has nothing to do with the Greek Philistines nor the later Arab invaders of Canaan.
Does this answer the question?