Oh boy, this is a very contested issue. They may or may not have the right to Jerusalem, depending on which side you take. This is best understood by looking at the history of Israel/Palestine. The basic history of the region is the following:
1. the original inhabitants are what could be called Palestinians (called in the bible Philistines, Amorites and other tribes)
2. The Hebrews arrived in the region from Egypt, where they were leading a pastoral life for a few hundred years, around the 13th century BC. They then established the kingdom of Israel (under kings Saul, David and Solomon), which later split into Judah and Israel. Jerusalem was founded by the Jews and was their historic capital.
3. Israel was conquered by the Assyrians in the 7th century BC, and many Israelites were deported to Mesopotamia, around Babylon. They were allowed to go back about a hundred years later, after the fall of the Assyrian empire. When they got back to Palestine, they found that some Israelites had mixed with Palestinians and had becomed what are called Samaritans, which the Isrealites didn't get along with.
4. Israel was conquered by the Romans in the latter half of the first century BC. The Jews revolted against the Roman occupation: this is known as the Jewish or Judeo-Roman wars. In 72 AD (I think), this rebellion was crushed and the unruly Jews were deported to all areas of the Roman Empire (Europe, North Africa, the Middle East).
5. For the next 1900 years, or almost, the great majority of the inhabitants of Palestine/Israel were not Jews but the original inhabitants- Palestinians. The Palestinians mixed in the 7th century with the Arabs who had conquered the Middle East. Most of them converted to Islam and adopted the Arab language. They are therefore nowadays considered Arabs, although most of them are probably of Palestinian descent.
6. In the 19th century, there was a movement called Zionism, whivch called for the existence of a separate state for the Jews, who had lived since their deporation from Israel in Europe, America and the Middle East. Most Zionists wanted the new state to be on the same territory as the original one, since Jews believe that the land was given to them by God. They started emigrating there, in small numbers at first.
7. After the 2nd world war, and after many Jews had been massacred in the Holocaust, the Allied powers decided that the Jews needed their own state. Britain controlled Palestine/Israel at the time and strongly encouraged massive Jewish migration. The state of Israel was founded in 1948 and immediately had to fight with its Arab neighbors, who felt outraged at Israel persecuting and deporting Palestinians. It has since adopted a militarist stance, persecuting Palestinians and defending itself against its Arab neighbors (war in 1967), and Arab terrorists who fight because of the Palestinians persecution.
It may be worth mentioning that Jerusalem is a holy place for Jews, due to references in the Bible, to muslims, due to Mohammed supposedly commencng his ascension to God there, and to Christians as well, becasue of Jesus' connection with Jerusalem.
In brief, Israel deserves Jerusalem because they controlled the territory for 1200 years in ancient times and also for the past 50 years. They also believe that they deserve it for religious reasons. The Palestinians deserve it because they were the original inhabitants, and also because they controlled for the past 2000 years, except for 2 brief periods: one since 1948, and the other before 72AD. They also believe that they desrve Jerusalem for religious reasons, due to its association with Mohhamed.
Hope this helps you. I know it's not an easy decision and I wouldn't actually advise you to say that one side or the other deserves Jerusalem.