Damn, I love that idea babalou! It's like Mario Party gone violent. Mario Bash, you might say.
I think the only games that are worthwhile to play online are games where it's really obvious you're playing against humans. Otherwise, it's just like you're playing against bots who use profanity. I think Advance Wars would make an interesting online offering. Any game where a player's personality comes through in their play style would work well. Racing games would not. Sports games mostly would not. Strategy games, that's where it's at, man.
I know this sounds stupid, but I think some type of online gambling game where you play for, I dunno, online time/credit might be... well, addictive, if somewhat shady..
Well, one cool online feature could be to let people trade gamesaves (anyone remember the dexdrive?), or trade downloadable mini-games a la Animal Crossing.
Hell, while I dreaming out loud... it would be awesome if there were some "game-maker" type software (like Fighter Maker for PS2) that let you make your own games which you could then share with the world. "MAKE YOUR GAMING DEBUT ON THE GAMECUBE NETWORK." Now that's a network I would call "DREAM ROAD". Or at the very least, it could let you download new textures, new sample/template characters, new music, etc.. Hell, even a BeatMania type game (a "music creater/mixer" type game, big in Europe, I think) that let you send your tracks to others, download new sound samples, instruments, etc.. Imagine competitions for where the winner could get his music incorporated into an actual game--maybe as the intro song, or a sample song, in the next Beatmania title.
And, of course, people could upload or download new levels made using the Level Editors which are included with many games (THPS, Timesplitters 2?, etc)