I think the hands down winner for 2001 is Grand Theft Auto III. It changed gaming forever and it was just the best game of that year. I can't tell you how many countless hours EVEN I spent playing! I just did random stuff not really knowing what I was doing and it was so much fun.
I also liked Super Smash Bro's Melee and I remember playing Gran Turismo 3: A Spec a lot because it was my cousins fav game. Being the typical guy he was, he taught me a ton about cars and showed me them on that game. Being the innocent little girl taking in everything he said, I actually listened and learned to like cars myself. I still don't have his knowledge and he was the better driver ALL the time. But he was nice and let me have a chance sometimes. Like I'd drive a car with 400hp and he'd drive a 49hp Fiat 500! Lol! But I remember this one time we each had the same cars and legit tried and so did I... and I beat him! I BEAT HIM! I couldn't tell if he was upset or proud I beat him from learning from him! Nonetheless he challenged me again and blew my doors off. But that one time.... I FELT SUPERIOR!
So great memories and those are my top three picks.