OK let’s break it down:
Rejecting arguments for Israel:
The Biblical claim – nobody except true Jewish and perhaps some Christian believers can accept this claim.
The Legal claim - the legitimacy of Balfour rests on the strength and interests of British imperialism at the time, not on any universally established rights. The UN decision of 1947 has more merit, but by then reflected facts on the ground rather than the original Zionist claim.
The Ancient Homeland – claim rests on a 3,000 year-old connection with the land, and most importantly on the existence of a sovereign Jewish entity in Palestine which came to an end around 2,000 years ago. However, there is zero precedent in any case in history of a claim based on such an ancient history being accepted. Imagine the reaction today if the Romanis of Europe decided to claim a state in Gujarat based on the fact they left there during the Middle Ages? If the land had been empty, then there would have been no problem. But while Zionist claims based on a sovereignty that last existed 2,000 years ago explains their focus on Palestine in particular as the location for their state, they cannot logically override the claims of the majority of the people inhabiting the land more recently.
Jewish Suffering – the strongest moral case for a Jewish state is the plight of the Jews throughout their history, and the belief that the best remedy for the so-called ‘Jewish problem’ is the Jewish people taking control of their own destiny through the creation of a Jewish state to sit as equals alongside the rest of the world’s modern nation-states. However, there is no logical way to argue that the relief of Jewish suffering in this way need be achieved at the expense of the Palestinians, since any suitable land available should suffice. And even if no empty land could be found, then the people to pay the price for the creation of such a state should lie somewhere between the Rhine and Volga, not in Palestine.
Tiny Israel vs. Expansive Arab Lands – it has been argued that since the Arabs inhabit vast lands from the Atlantic to the Gulf, they should get over it and accept Jewish sovereignty on a tiny piece of that land. This is an argument with some merit – however, there are obvious ways to go about requesting such land, and it is clear that, petitions to Sultan Abdulhamid and the Faysal-Weizmann negotiations notwithstanding, the Zionists were determined to establish their state no matter what the native Arabs agreed to.
There is no Palestinian people – Golda Meir’s famous line has the merit of being based on an element of truth in that the Arabs of Palestine were late in adopting a nationalist ideology through which to express their claim to the land. However, the fact that the Zionists found the Arabs in such a politically immature existence when they arrived should not have any moral bearing on the fact that they imposed their state on the locals without their consent, Palestinian nation or no.
Population Exchange – the argument goes that since a similar number of Jews were forced from Arab and Muslim lands following 1948, the Palestinians should accept their plight in terms of a population exchange. However, there is no reason Palestinian Arabs should be forced to pay for the actions of the other Arab states in their treatment of their Jewish population.
In rejecting Zionist claims based on the above propositions, the Palestinians were not doing anything abnormal according to what we know about how humans generally respond to the attempt of foreigners to impose themselves on a native population coercively (through hard or soft power). The fact that a host of factors have since led the Palestinians to fall to the sickness of antisemitism, conspiracy mongering, and indiscriminate terrorism, has no bearing on the legitimacy of their original and understandable rejection of Zionism.
A precedential case for Israel
Despite the above, a case can be made for the legitimacy of Israel based on precedent – Zionism as a legitimate national movement born from the same world and playing by the same rules as the nationalism of the rest of Europe (and increasingly the world), and the many cases where modern states have been born out of similar or worse circumstances than Israel, without receiving half the scrutiny Israel gets for being born in ‘original sin’.
The context – the Zionists were men and women who grew up in a world in a total flux, with all the traditional forms of political order falling apart around them, borders being redrawn, populations on the move, and with every ethnic group from the Alps to the Caucasus attempting by whatever means to achieve some measure of political autonomy. In this predicament, European Jews had a few choices. They could stay and either assimilate (mostly possible in Western Europe) or keep their heads down in their ghettos and try to weather the storm (Central and Eastern Europe, with tragic consequences). They could leave, to America or elsewhere in the West (which most did). Or they could try to negotiate the mess on equal terms with the other players by adopting nationalism - it would have been unnatural for Jews not to have been drawn to nationalism in such a situation. Therefore, there is nothing especially unique about Zionism that distinguishes it apart from the other forms of nationalism adopted by various European groups at the same time, apart from the fact that there was no obvious European territory up for grabs for these Jews to focus their attention on.
Right of conquest - If the idea that modern states cannot exist on the basis of conquest and settlement was applied fairly, dozens of states in the world would have to be disbanded. Israelis understandably get irked when being lectured by Americans, Australians and the likes on their treatment of the native Palestinians. Closer to home, the origins of the modern state of Turkey go back to the medieval conquest of Anatolia by the Seljuk Turks and subsequent Turkish migration from Central Asia, and the state was achieved by the ethnic cleansing and/or murder of a couple of million Armenian, Greek and Assyrian Christians (as with the Zionist project, there were mitigating circumstances, though probably less vital). The problem here for the Zionists is not that they dispossessed the Palestinians, but that they didn’t do it earlier, at a time when they would have got away with it without a fuss, and that they didn’t do it thoroughly enough, so as to leave no Palestinians around to try and get even with them. With this in mind, attempts to argue for the illegitimacy of Israel based on the supposed uniqueness of its original sin cannot be accepted given the number of other states born of similar or worse crimes, unless the argument rests on an entirely arbitrarily chosen place in time only after which the creation of such states became unacceptable.
National self-determination – unfortunately for the Palestinians, the fact is that, for the reasons outlined above, a community with its own distinct characteristics which fulfills all the elements of what we commonly accept to constitute a 'nation' now exists in Palestine. Attempts by the Palestinians to dismiss the Jews as simply a religious group, as the original PLO charter does, or as some type of modern-day Crusaders, something very common in Palestinian Arab discourse, are no less contemptuous than Golda Meir’s denial of the existence of the Palestinians. And in an international system based on the principle of national self-determination, in which new states routinely come into existence based on ethnic separatism, the Jews of Israel are no less deserving to this right than any other nation.