I can talk a little about Adolf Eichmann specifically. Israeli law says they did. But they also claimed it under other legal principles.

The Israelis claimed "universal jurisdiction," which asserts that certain crimes are committed against humanity as a whole, and therefore can be prosecuted anywhere. And of course Israel has, since its founding, considered itself as the legacy of those killed in the Holocaust, which gave another basis for their claim of jurisdiction.

Universal jurisdiction is just an idea, though; there wasn't any kind of international treaty backing it. Eichmann's defense attorney made this a central point of his defense, arguing that since Israel didn't exist until 1948, they had no right to try Eichmann for prior crimes. It's worth mentioning that the Genocide Convention, which was ratified in 1951, does not automatically grant universal jurisdiction.

So while there wasn't really any codified legal basis for the trial, no country really argued about it, because it's Adolf Eichmann. Argentina kicked up a bit of a fuss, but that was more about the fact that Israeli commandos captured Eichmann in Argentina without Argentinian knowledge or involvment.

Hannah Arendt's Eichmann in Jerusalem is the place to start on this, if you want to learn more.

It's more of a legal question, and it's a very debated topic.

For starters, jurisdiction means the power to say what the law is and it comes from the latin "juris dictio". So the question of Isreal having or not having jurisdiction depends on whether universal jurisdiction is a thing or not. That is, can countries prosecute crimes committed outside their own borders, crimes to which the State has no apparent connection or interest in prosecuting.

The second big obstacle in accepting Israeli jurisdiction is the special nature of criminal law. Criminal law has unique principles (natural judge, irrectroactivity of the law, legality, etc) that are in place to protect defendants. These obstacles were also faced by the Nuremberg trials (or by any international criminal court, although presently there are better arguments against those objections).

The fact that Isreal did not exist until 1948 is not a particularly good argument, since until the International Criminal Court, all international criminal courts were ad hoc: created after the fact to address the crimes committed during the passed conflict, like the courts for Rwanda or the former Yugoslavia, and dissolved after there were no more crimes to investigate.

In my opinion, the biggest objection to the Nuremberg trails, and by extension to the Israeli trials against Nazis, is the lack of positive written law criminalizing genocide. Before 1945, there was no universal treaty or agreement on what genocide or crimes against humanity meant, nor how these two crimes would be prosecuted and punished.

Both the Nuremberg trials and the Israeli ones supported their claims on the idea that there existed a customary international law that punished those crimes: there was a dated and established practice that incurring in said actions constituted a crime. Nowadays, this argument is stronger since we can find a lot of treaties and UN resolutions that define what international crimes are. Of course, for the crimes commited during WWII, at the time none of these instruments existed.

In summary, the problems the Israeli trials faced were two-fold: whether they had the power to say what the law is (jurisdiction), and what law is applicable (considering that there was no written law on the subject during WWII).