I'm going to essentially summarize this article from 1961, because I think it does a good job of describing the legal perspectives. I'll add in other things as necessary, but I want people to know what I'm crediting. I don't want to plagiarize its arguments, so I'm giving it credit :).

Few reasons.

1. The Israelis are the ones who caught him. He was hiding out in Argentina, and Israeli agents kidnapped him.

2. No one was going to be able to tell the Israelis, the Jewish state, that they should hand over a Nazi war criminal for trial anywhere else. They wanted to do it themselves.

3. Israel claimed jurisdiction. I'll explain that further below.

So to charge Eichmann in Israel, there had to be two prospects fulfilled. Israel had to have a law that it could prosecute Eichmann under and to charge him with, and it had to ensure it wasn't violating international law.

On the question of Israeli law, Israel had passed a law in 1950 called the "Nazis and Nazi Collaborators Punishment Law". This said that any person who helped the Nazis or collaborated with them was liable to receive the death penalty, and can be given a lesser sentence if necessary. The law itself says Israel has jurisdiction even if the actions were committed in another country. So under Israeli law, this was legal. But then there's the other question, about who has jurisdiction, right? I mean, just because Israel says it has jurisdiction, does that mean it really does? That's where international law and legal norms come into place.

On the question of international law, there's a few issues...

Eichmann was abducted from another sovereign state. This is against international legal principles, which say that such a violation by foreign agents is a tort. If volunteers had, without Israeli government support, gone ahead and abducted him and then turned him over to Israel, that's pretty much the only way Israel wouldn't have necessarily broken the law there. Argentina could theoretically have demanded the return of Eichmann, but Israel could've argued that it had the right to deny this because Argentina had previously (years before) said to the world that it would turn over all Nazis and collaborators. This was a tenuous argument though, because of the difficulty of actually arguing that this pledge back in the 1940s applied in 1960, or was binding.

But this dispute over Argentina v. Israel was settled regardless, by the UN Security Council (UNSC), when the UNSC suggested that Argentina shouldn't request the return of Eichmann and instead suggested it should accept just an apology. Only Argentina could've requested he be turned over based on the method he was abducted, but Argentina ended up not making such a formal request, at least in part because it didn't get that backing from the rest of the world.

Then there's the question of Eichmann's nationality. He was German by citizenship, but also got citizenship under a fake name as a stateless person with papers issued by the Vatican. The German Federal Republic could have claimed it had the right to prosecute him, since he was a citizen, but it refused to do so and instead welcomed Israel's case against Eichmann. It even offered to provide and gather evidence if necessary. And besides, Eichmann may have vacated German citizenship's protections by claiming a stateless person's passport.

The next issue is whether or not Israel could try him under a law passed in 1950, when the crimes were committed before the law was passed. This seems to me to be a straightforward question, considering that the Nazis were tried for crimes against humanity at Nuremberg despite the laws to try them not necessarily existing yet.

There's another question not covered by the original article that I think you're talking about too. Why didn't someone suggest an international tribunal? Israel could claim jurisdiction under international law because the world essentially accepted it, and also under the claim of "universal jurisdiction", which basically says that war crimes and crimes against humanity (particularly those that take place across national borders) don't have to be tried in the original jurisdiction it was in. Any country can try the person for war crimes and crimes against humanity. If another country had claimed Eichmann, they could've argued over who deserved it. But no one did. Nor did anyone really suggest taking away Eichmann from Israel for an international trial, as I mentioned at the start. As such, without an act of the UN Security Council or some other binding international body, there was no tribunal to try him and no one else who requested to do so.

So it wasn't necessarily that there wasn't the possibility of it. Simply put, it came down to the fact that Israel had the laws, the right, and the motivation to try him, and no one was going to take Eichmann away from it.