I've seen very little of GTA3, but I noted that lots of people had fun with that, as well as got addicted to FFX, so I'll throw those in.
In addition, I'd like to throw Diablo II and/or it's expansion pack, Lord of Destruction, into the mix, because the expansion is SO worth getting to add onto the game. I got addicted to that this year...
---=---:---=--- [Look up a line] Wow, that belt looks like crap... I'll take it and carry it around for years and see if people can take it away from me... What? What was that? You mean even THIS belt doesn't matter! What? I'm insulted! Get away from me you... you... defiler!
As a person who only ever buys sports games, I gotta go with a special nod to Metal Gear Solid 2, since it's the first game of its kind that I've have ever felt that "Oooooh I gotta have that" feeling about.
On the sports side of things, I'll go with NHL 2002, although I'm not too keen on the Free Agency feature in Franchise mode. Couldn't you just copy the way Madden does it? That one is the best.
I also love the new play on FIFA 2002 and NBA Live 2002. Much different an better than the 2001 versions.
The best new thing on an old game for me is the recruiting on NCAA Football 2002. The fact that the game is now capable of naming all the players and that you can see guys you recruited but failed to get play against you during the following seasons kicks ass. Plus, the gameplay still kicks ass. In fact if I had to only keep one game, that would be it. But you'd also have to pry NHL 2002 from my cold dead fingers.
My two obvious picks here are FFX and GTA3. My third one that hasn't been mentioned would be Max Payne. It was definitely one of the coolest games for the PC with Bullet Time shooting and a lot of crazy action.
Worst game of 2001? Probably the interactive Survivor game.
I have hardly played Smackdown 3 alone, which sucks since *I'm* the one that bought it, but it is the one game I have that my friends will always want to play whenever we have a few people together. It's great as a two or more player game, but I agree that as a 1 player game it doesn't have much replay value.