First off, let me say that I love my Gamecube. I it love just the same as my Dreamcast. That said, Dreamcast is all but dead except for the underground community. So what can Nintendo do to survive against Sony and the Microsoft juggernaut?
As the Nintendo CEO, the first thing I would do is to call Rockstar Games and promise them the world. Grand Theft Auto is probably the best game ever save for Zelda or Super Mario. Notice which 2 I put in the same category? Two Nintendo games. But those games have for the most part, lost their newness. Sure, the last Zelda game was incredible, and the next one looks to be a masterpiece, but they don't compare with the "coolness" of GTA. Microsoft understood this and picked up the phone to Rockstar. Its time to do the same thing Nintendo.
Second, their next system must have the ability to play DVDs. No exception. The reason is not only the double feature of being able to play movies and games, but also because of the size limitation. Right now, their game discs are smaller in size, but higher in tech than the PS2. The Gamecube is higher on the tech chain that the PS2...but to be honest, the PS2 has GTA and several other cool games. The xbox has Halo and GTA.
Third, Nintendo needs to get online...and make it free. Thats right, free. Xbox live has been a huge success, and now its time for Nintendo to fall in line. But they can sell more games to go online with if they make it free. They will trounce Xbox Live if they get the good enough games with the free online service.
As far as the Gamecube goes, you can't beat the price. Its 2/3s the cost of either of the other systems. Also, Nintendo is on the right track apparently with their Nintendo DS, the next generation Gameboy. I've seen it, and I couldn't believe my eyes. Metroid Prime looked amazing on it. Plus, you can play it wirelessly with multiple players.
I'm sure Nintendo won't take my advice, but maybe they'll wise up themselves before its time to roll out the next gen console.
"Now You're Playing With Power!"
Update 20 minutes later:
Apparently, the next gen console from Nintendo is codenamed "Revolution". All you have to do is look up "Nintendo Revolution" in google and you'll see the details. Sounds interesting...