I love trying to convince people to buy Nintendo products. I've sold at least one Gamecube so far with my convincing, let's see if I can do it again.
Like couchmonkey said, RPGs are very present on the Playstation. It has more than you could ever play, but I'm guessing that you're more interested in playing game than trying your hands at an RPG, which are generally very long and story oriented. Nintendo puts gameplay first, following it with story, graphics, and other details (which are usually top-notch, regardless).
Xbox has more first person shooters than you could ever play. You could play most of its quality games on the computer, but it has got some fun games. Not my taste, definitely, and the selection of good games isn't too diverse. If you're into war games or first person shooter games and don't like computer gaming, you're probably best fit for Xbox.
However, Nintendo.
In my opinion, Nintendo has by far the highest quality games. True, most of them are appropriate for all audiences, but they are not exclusively for children. Mario is tons of fun for anyone who plays it, games like Pikmin are fun and challenging, and party games like Mario Kart and Super Smash Brothers Melee are a blast. My friends, even my Xbox ownin' friend and my Final Fantasy fan friend, love SSBM, and everything my friend murk and I have on our Gamecubes. It's a ton of fun to play all the games with people you know, very laid back, fun, and cool.
What I enjoy most about Nintendo, though, is the one player experience. I just finished playing Metroid Prime 2: Echoes and it, like the first, is an excellent, challenging, involving game. Even my roommate, who is a very casual gamer, got sucked into it, saying "Uh oh, I'm saving a game. This can't be a good sign."
Other great one player games are Pikmin 2 (can't speak for this first, but the sequel is amazing), which is an interesting strategy/adventure game, Wind Waker, a great adventure game with stylistic graphics (very colorful and bright, but impressive, so long as one is not shallow about that sort of thing), and F-Zero GX, an intense, futuristic racing game (has multiplayer, but the single player is much more fun and challenging).
Many other games, too, but I thought that was a pretty diverse selection, deciding between three games.
Fun party games: Super Smash Brothers Melee (fun to play, and it can get very deep in terms of gameplay mechanics), Mario Kart (great, fun racing), Super Monkey Ball 2 (have only played the first, which was awesome, but I hear this is even better. Monkeys in plastic balls, more fun than it sounds, I promise), and you might want to check out the upcoming Mario Party 6. Four was below the usual standard by quite a lot (though my one friend can never stop playing it whenever we get together. Different strokes for different folks, I guess), and five was good, but this one looks like it has promise. Pikmin 2 is two player, but my friend and I were laughing hysterically while playing it, lots of fun.
Those are all Nintendo games (excluding Monkey Ball, which is Sega), or Nintendo franchises made by other companies with Nintendo supervising (F-Zero GX was also Sega). Good third party games are on the gamecube, though, like Viewtiful Joe (the sequel came out recently, actually) by Capcom. I'm not too aware of all of them, though, since I usually buy games more closely affiliated with Nintendo, and that takes all my money.
Nintendo is strongly oriented toward a broad fanbase, including children and adults, for the most part. The games are definitely fun, no doubt about that, and any tiku tiku tiku! nonsense you hear is mainstream media talking (the same media which makes Britney Spears and bad rappers top the charts, musically. The media is not to be trusted, especially with video games), and I expect you're mature enough to ignore the rumors that Gamecube undermines your manliness. Quite the opposite, really

Hope you make the right choice, and good work, doing research. That's always admirable.