Funny given the comments under the video, as Ice Man would also have been my choice for this debate.
That said Wolverine is in a league of his own as few comic characters have taken off quite as well as he has, and endured quite as long. While numerous X-men have come close to his popularity, I don't think any of them have maintained it for nearly as long, and I'd even go so far as to say he's probably one of the most recognized Marvel characters in general, he comes to mind almost immediatly when Marvel comes up alongside other characters like Spider Man.
If I had to defeat "Wolverine is the coolest" X-man arguement, I'd argue that by becoming so iconic and mainstream he effectively lost the abillity to be cool. To be truely cool requires a sort of fringe prescence that he doesn't have any more.
Ice Man was kind of the original joker/outsider of the group, who had the whole cocky "does not play well with others" thing going. What's more he winds up being on and off the team for various reasons, either the team having issues with him, or vice versa. When Ice Man shows up though you generally know things are going to get awesome... in part because he's one of the most powerful mutants there is (one of the few that was ranked as Omega class or whatever it was) even if it's not always as obvious as say Jean Grey (with her own powers, not considering Pheonix possession) because he so rarely uses little things like complete and total control of water molecules and the abillity to freeze them. It rarely comes up, but in a few occasions where he used his potential it was so god awful nasty that I'd imagine he's written out because writers just don't know what to do with the character. I think it was established at one point that even his fear of melting is irrelevent since he can pretty much reform from ambient moisture.
It's been quite a while, but basically there is the old "superman throws all of his problems into the sun" thing potentially at hand, since unless it's a divine class threat if Ice Man ever decids to remember he could say freeze all the water in someone's body (or just their brain) instead of making constructs it's pretty much over.
Overall, I think Ice Man is pretty much the "James Dean" of the crew.