I've spent years wondering about these things. ...probably longer than the question is worth, to be honest. The only reason I could see why I would want to know what "intelligence" is is so that I can call someone "intelligent" or "not". It's kind of stupid, once i thought about it. If someone's good at making distinctions, then it doesn't matter if that's really the root of intelligence--I'll talk to them whenever I think there's a difference between two similar things that I'm not sure if I'm getting. And whenever I have a brand new problem, I would want to talk to the guy who comes up with creative solutions, whether he's "really" intelligent or not.
It could be so many things. Every 5-7 months or so (averaging over maybe 4 years), something new would come to mind, and I would think "ahh I forgot about that. I think a person has to be smarter to do X than to do [whatever was the past best criteria]... so maybe that's really a more fundamental kind of intelligence." It happened so many times that I don't even remember them all (to name a few, though: recognizing the differences in similar concepts. Seeing the similarities between disparate concepts. Being able to explain their ideas in such a way that many people truly understand them. Following a single thought/belief's implications through to see what would happen if the plan were implemented or the belief adopted. Recognizing contradictions within a system of ideas. Being able to get inside of a system of ideas and fairly judging whether they were consistent... the list goes on and on and on).
But it's not like a linear list. Someone can be good at spotting contradictions and bad and finding similarities (Ti vs Ni). Someone can be good at explaining complicated ideas, but have no idea where they lead (Si vs Ne). Some people have a big list of these that they can do well, but I've never met anyone who could do them all. I eventually just decided that I didn't care what made someone intelligent, and shortly after recognized that even if I did know, it would probably just make me look down on some segment of unintelligent people just because they fit into the category... which then of course closes you off the subsets of intelligence that they do have, and things just sort of go downhill from there.
Maybe that's just my body's way of leveling out the cognitive dissonance associated with failing to come up with anything... but I sorta think it was just a frustrating waste of time, thinking back on it now