My favorite old system has to be the SNES. at least a third of the games that are now considered classics were for that system. There was Super Mario World 1 & 2, Star Fox, EarthBound, Final Fantasy 4 and 6, Chrono Trigger, The Secret of Mana, Super Mario RPG, The Donkey Kong Country trilogy, The first two Earthworm Jim games, The original Mario Kart, and my two favorites: A Link to the Past and Super Metroid, in that order.
It also was host to E.V.O., which I consider to have the best portrayal of dinosaurs in any video game ever. But it's hard to appreciate if you're a creationist.
Other than that, my favorite old systems are the N64, and the PS1, although I don't actually have one--all the PS1 games I've played were played on my PS2. Metal Gear Solid is my favorite PS1 game, but I also like Castlevania: Syphony of the Night and Final Fantasy VII (Which I actually only have for the PC since I didn't have a PS yet when I got it.) The N64 had Banjo-Kazooie, Conker's Bad Fur Day, Rayman 2, and the best 3-D game ever: The Legend of Zelda: Ocarina of Time.
Ironically, my favorite game for the original NES isn't any of the really famous ones like Super Mario Bros. 3 or either of the first two Zelda games. It's a game called StarTropics that for some reason never got to be very well-known. I'm kind of surprised that it didn't--there's no reason why it shouldn't have.