Israelis believe they have a historical and religious claim to the land on the eastern shore of the Mediterranean Sea based on biblical narratives that link the Jewish people to the region, particularly as the ancestral homeland of the Jewish people. This belief is reinforced by the long history of Jewish presence in the area, as well as the Zionist movement in the late 19th and early 20th centuries, which sought to establish a Jewish state in response to widespread persecution. The establishment of the State of Israel in 1948 further solidified this claim in the eyes of many Israelis. Additionally, historical events, such as the Holocaust, have shaped a sense of urgency for a secure homeland.
The Israelis claim historical and religious connections to the land, dating back thousands of years to ancient times. They believe that the land was promised to them by God and that they have a right to live there as a homeland for the Jewish people. This claim is supported by archaeological evidence and religious texts.
Jerusalem
Yes, they do. See the related question on why that might be.
Both claim the entire Southwest Levant, termed by the Jews as "Eretz Yisrael - The Land of Israel" and by the Arabs as "Belaad Filastin - The Country of Palestine", as their homeland.
A discussion and explanation of Jesus claim that if you believe in him you will never die.
Historic Palestine or the Land of Israel. (Both are regional names for the same piece of land, roughly.)
It depends on how they make that claim. Bedouin and Druze claim that Israel belongs to them as much as it belongs to the Jews and proudly call themselves Bedouin Israelis and Druze Israelis. A number of Arabs who formerly lived in what is now Israel or the Occupied Territories claim that the Jewish State is illegitimate and demand that country be returned to them for administration. These individuals are called Palestinians. (Admittedly, a number of Palestinians are willing to allow Israel to exist outside of the Occupied Territories, but there is no difference in naming between the Two-Staters and the One-Staters.)
Islamic
Muslim
the Turks and the Greeks are disputing claim over Cyprus
Arabs believe they have a right to claim land for exactly the same reasons that all other people believe that they have a right to claim land. They feel a right to the land that they live on since they live there and their ancestors have lived there.
The Old City of Jerusalem is known for its sacred sites in the Jewish, Muslim, and Christian faith, solidifying each group's historical claim to the land. Both the Palestinians and Israelis want Jerusalem as their capital.