If nothing else, I'd name it a classic for being the first Spider-Man cartoon that had really good animation. It's appalling to me how many Marvel cartoons over the years haven't. I think it's also the one that pulls off the same trick as BTAS, which is that it successfully cherry-picks elements it likes from all that came before it and turns it into something good in its own right.
In the end, I probably agree with most that SSM is a bit too early to call a classic just yet, but I would agree that it's also well on its way there.
I don't think the definitive ending (so far) for Avatar should have much bearing on whether it's a classic or not. Cowboy Bebop in animation and The Prisoner in live-action are both finite stories (much shorter than Avatar, even), and neither one seems to be in any danger of being removed from "classics" lists. MonkeyFunk also listed a few other examples of finite series that are already animation classics.
For that matter, you could argue that all TV series are, by definition, "finite" because they all end eventually, as do movies.
At this point, I would probably single out the second season of Avatar as a classic, simply because it's one of the finest single seasons of television I've ever seen, animated OR live-action. The fact that I didn't find the third season quite as compelling as the first two (at least until after the midway point) is also a strike against calling all of Avatar classic. That may strike some as odd, but I wouldn't really say so. 24, The West Wing, and the new Battlestar Galactica are live-action examples with extreme highs at their best and extreme lows at their worst, and averaging out the entire series would seem to penalize the highs unfairly.
Organizations like the BFI are collectives of individuals. If an overwhelming majority of them declares that something is a classic, the odds that they are all wrong become much smaller. It's not zero ("if a thousand people say a foolish thing, it is still a foolish thing"), but it's greatly reduced.
Popularity is also a poor gauge for classic status for a number of other reasons. Some works were groundbreaking at the time and set a foundation for what followed, but are hard to enjoy strictly on their own merits today. Metropolis is one example -- I would definitely call it a classic, but I would not call it popular by any stretch. Most modern audiences wouldn't even get past the fact that it's black-and-white and silent. The original Jazz Singer is probably another example. There's also accessibility. Even though Charlie Chaplin or Buster Keaton were classic comedians and their films are still watchable and funny today, but most people today probably haven't even seen their films. They aren't "popular," but you're in for a long, hard fight if you try to claim that they aren't "classic."
Besides, if critics can be ignored because they could be wrong and, as they say, "everyone's a critic," then there's not much sense for anybody to talk about any films or TV shows at all, is there?
Of the shows in the past 5 years, I'd say that The Simpsons has probably already made it to classic status, and Avatar with the caveat listed above (which I reserve the right to change my mind about after I sit down and watch all of book 3 consecutively). In movies, I'd name Spirited Away and The Incredibles as classics since the last thread, since both managed to really push the boundaries both in technical and storytelling terms.
-- Ed