I am super surprized this thread has lasted this long, so i feel compelled to add my 2¢
-home brew gamecube emulation will NEVER come to a playable level on the 3DS (as said by many here only now are gc/wii games on pc emulators able to play at normal speeds, and only on the most powerful computers.
-It could be veagly possible that Nintedo develops a Virtual console for the 3DS, but even then I would find it hard to believe it could do GC games unless the games were overhauled and scaled down to better suit a smaller resolution.
Lets look at Nintendo's history of reusing old games and using emulation on newer handhelds.
The GBC had a few NES titles
The GBA focused on some SNES and NES titles and if i recall right had quite a few of the better ones
The DS has focused on mostly on SNES titles but there has be one or 2 N64 title we have seen, like mario64, how ever mario64DS was from what I understand it basicly a full remake of the game not emulated as it was on the wii's virtual console.
Home-brew emulators to the above systems also for the majority follow the same trend when it comes the power of the system.
Following this trend, I'd say its a safe bet that this generation, the 3DS will focus on a lot of N64 titles(like Zelda OOT) where the DS did not/could not. Which may allow home-brew developers to create a N64 emulator. There are many N64 great N64 titles that the DS never saw, it would have been in Nintendo's interest to only invest enough hardware into the 3DS to play titles unique to it and up to N64, jumping the hardware past that wouldn't be as financially effective.
The DS had a hard time in some cases running SNES home-brew emulators, and it was later improved in the DSI, which if I remember correctly at that time could finally run home-brew GBA emulators.
Conclusion:
-GBA and prior - definitely
-N64 - quite probable
-Game Cube -SLIM CHANCE(titles in the future may be seen if they are remade/overhauled as was Mario64 for DS)
Depending on the generation of titles Nintendo re-releases form a older system for a current system, dictates that it can also be emulated, if Nintendo dose not release those titles on the current system then they generally can not. Simply look at the list of upcoming games, and you will be able to tell if it can do it
http://en.wikipedia.org/wiki/List_of_Nintendo_3DS_games