I didn't own a PSOne until a friend gave me his right after he got his PS2. Excitedly, I went out and searched the web for the highest rated PSX games in order to know what to spend my money on. These are the ones I have found worth buying:
Tomorrow Never Dies (absolutely sucks, but being a Bond fan, I really enjoyed it)
Oddworld: Abe's Oddyssey
Oddworld: Abe's Exodus (both are good fun and challenging side-scrollers released in an age when the side-scoller was obsolete, which adds to the coolness)
Metal Gear Solid (there is a boxed set released earlier this year containing the first 3 MGS games, 1 for PSX, and 2 and 3 for PS2 for around $30, definitely worth it)
Tony Hawk's Pro Boarder (good mindless fun, gets old eventually, good game in its time, pretty dated now though)
Spider-Man and Spider-Man 2 (being a long time fan of the Spider-Man comics, this game was a dream come true, the web slinging was fun, and it was chock full of references to the comics and Marvel easter eggs. The second one wasn't near as good, but still fun. Predates the movies)
Resident Evil games (old news by now, but I still think they are a lot of fun to play. Great classic B movie feel to them)
Episode I: The Phantom Menace (piece of crap actually, horrible graphics, but it was kind of fun going around with a lightsaber and slashing up droids, levels where you play as Jar Jar are unforgivable though. I heard Jedi Power Battles was another fun TPM inspired lightsaber slash em up game, but I have heard from other sources that it sucks some serious balls)
The City of Lost Children (I just thought it was great that that movie had a game based on it, bought the game and never made it out of the first room, uh, so.. yeah, I guess I can't really recommend it)
Skull Monkeys (I obtained this one from somewhere, but I still have never played it, I really liked "The Neverhood" for PC back when it was released, so obviously I needed to own its sequel. Not quite sure why I never played it)
There are plenty of "must play" games on the system that I have never played (Final Fantasy and a few that PSX fans used to worship back in the day). I was an N64 and PC guy back in those days, I always felt, and still do feel that the original PlayStation was pure shit. The graphics were way behind its time, load times sucked and this problem only grew more severe as your system aged. N64 games looked a zillions times better in comparison, though the N64 suffered from a very small library of games, the few good games it did have were REALLY good. The PS2 on the other hand, tough still under powered compared to its contemporaries, was one of the most amazing systems ever made IMHO.