It wasn't so much a UN thing as a British thing. They thought "If we pander to these Jews maybe they will influence their governments to help us win WWI" and then screwed over the Jews and Arabs (who they had a separate deal with) as soon as the war was over. On the OP I think you are going to need a pre-1900 POD because by then the Zionists were pretty well set on Israel/Palestine.
Yeah, there were quite a few Jewish immigrants to Palestine before 1948* -- it's not like they all arrived on one boat.
Cyprus is an interesting idea, though. It would lead to a radically different Israel, as you wouldn't have anything like the Six Day War (if Israel was an island nation, they'd have even less ability to get other people to recognize their right to expand). The Greeks and Turks would react negatively, of course, because obviously the current situation there shows that they don't want to leave. It could be an interesting TL, though.
I'm sure that Arab-American, Arab-British, and Arab-Jewish relations would all be better, though. The Middle East would still be a messed up place but not so bad, I think. It'd be interesting to see how Palestine would develop without being occupied, and also how countries like Egypt and Syria would change their policies (less military buildup, for example -- but also it's possible Nasser could've never come to power without the Israeli threat).
*Along with some pretty sketchy British policies such as allowing the Jews to have guns but not the Arabs...