The biggest thing I disliked about my gamecube was the lack of support.
First, some big developers started withdrawing from NGC developement. This meant that lots of great games on my ps2 and the xbox wouldn't make it over.
Then, the knock on effect was that, whereas we waited about 1-2 weeks for every new ps2/xbox title, I remember having to wait 1-2 months for the next NGC title (ALWAYS the same old games whenever I checked).
The thing that sells consoles isn't the hardware spec...it's the games....and the serious lack of games (esp "grown up" / adult themed ones) really started to annoy me after a while.
Yes there was Rogue Squadron II & III, resident evil 4, etc etc.....but there still simply wasn't enough c.f the other contenders.
On top of this, ninty simply stopped promoting it's own product. I can't even remember seeing any adverts for it anywhere (unlike sony/m$).
Walking into Game/Gamestation you'd see a whole wall of xbox games and 2 walls of ps2 games. I'd then have to sheepishly walk past, over to the back of the shop, where there was a lone small cupboard with NGC games (almost labelled as the "you're too poor and embarrassing to shop with everyone else" cupboard).
How many times did I walk into my local whsmiths only to find that the game section simply didn't contain
any NGC games at all?
It's as if a lot of shops simply ignored the ngc completely in favour of the xbox/ps2
I hope they don't make the same mistake with the Wii.....yes it's fun, but ninty & others shouldn't make the mistake of assuming we're all 12 yo.
Games like Red Steel and Heatseeker are
good signs imo that this time we'll get a "grown up" console

.
The cessation of NGC support isn't too bad, because we can still use a Wii to play the same games (and it's pretty cheap at £180 as it is).
Again, the console was good, yes, it was the lack of support (esp from developers) that sucked.
What I'd like to see on the Wii to stop it "doing a gamecube" is:
GTA (if it's on the 360 and (eventually) ps3, why not on the Wii?)
Pro Evo (ditto)
Serious racing sim anyone (a la "GT5/Forza")
Wipeout clone (before anyone says, F-Zero is a completely different style of game imo......no weapons make it completely different for a start) [or maybe the next F-zero should include weapons

]
Fortunately, I predict that the new control system will have developers flocking to the Wii, hopefully broadening the spread of genres massively (and enticing shops to at least stock them).