The main claim of the Israeli Jews, even secular Jews who do not really practise Judaism, to the land of Israel is a religious one - their God promised the land to Abraham and his descendants. In this, they are perversely aided by the Palestinians themselves, whether as Muslims or Christians, because they worship the same God and give credence to this claim. A secondary claim is that Palestine was once the land of the Jews and they should rightfully be allowed to return.
The main claim of the Palestinians is that this has been their traditional land for at least two thousand years. The likelihood that the Palestinians could be descended from the former Canaanite and Philistine inhabitants would push this period of continuous occupation thousands of years back into antiquity.
The claim that it is the Palestinians who immigrated into the land in recent times is demonstrated to be false by the British census taken following the end of the First World War, in which the British classified the 800,000 inhabitants of Palestine according to religion: 650,000 Muslims, 80,000 Christians and only 60,000 Jews. The Palestinians can claim to be descendants of the pre-biblical occupants of Israel. Similarly, there never was a League of Nations condition, implied or otherwise, on the British Mandate that any part of the Mandate be made independent, either as a Jewish or Palestinian state. After the Second world War, the British government determined to create a unitary, democratic state (Palestine) to the west of the River Jordan, but Zionist terrorism, including the assassination of the British representative and the bombing of the King David Hotel, resulted in Britain retreating from this proposal.
Involved in all of these claims, the main determinant is power. The nation of Israel is militarily strong and can impose its will. The Palestine Territories are weak and for them, merely to return to the 1967 borders would be a successful outcome and a sound reason to accept a peaceful solution. Moderate Palestinians now limit their claims to the territory within those borders.
The Jewish claim to Israel is that Jews have lived on this land without interruption for over 3000 years. That this land is their ancestral land. Religious Jews have the additional claim that the land was promised to the Jews by HaShem.
The people who now refer to themselves as Palestinian (prior to 1967, the only people who called themselves were Jews, the Arabs referred to themselves as Syrian) are made up of a combination of descendants of Jews who converted to Christianity and Islam in ancient times, Syrians, Lebanese, and Egyptians. A large percentage of these people did not enter Israel until after 1920 when the British put no limit on Arab immigration into the territories.
The country that is now called Jordan was original meant to be a part of the Jewish homeland, as per the League of Nations. However, the British gave that land to the Arabs (in violation of the mandate handed to them but with the agreement of the Jews) as a Palestinian Arab homeland. The remaining land was then further divided but there are no 1967 borders, those were armistice lines.
The Arabs who chose to not support the attacking Arab countries in 1948 went on to become full Israeli citizens with all the rights citizenship entails.