Originally I though Super Wii or Wii 2.0 would be good names. However now I think Nintendo Wii Advanced might be good. It keeps the Wii brand but Advanced I think gets across the idea that this is a new more powerful and simply more advanced system compared to the original Wii. It plays off the difference between the Game Boy and Game Boy Advance.

WII HD or nintendo U

Nintendo Pad
or
Wii Pad

Nintendo's naming conventions tend to fall into two categories, naming a system in a way that has an obvious association to what the system is and those with a more arbitrary association.

Obvious: Nintendo Entertainment System (NES), Super Nintendo Entertainment System (SNES), Nintendo 64 (N64), Gamecube (GCN), Virtual Boy, Gameboy Advance (GBA), Nintendo DS (NDS) and the Nintendo 3DS (3DS).

Arbitrary: Gameboy (GB), Wii, Wii U


For very understandable reasons Nintendo wants to keep the Wii name which I think makes it more difficult to come up with a meaningful name. The Wii HD has value being that this will be Nintendo's first HD console but I don't think Nintendo wants to put too much emphasis on that. Since their focus is on the controller, there are obvious name choices which are absolutely terrible like Wii Stream, Wii Touch and Wii View.

One thing that might work is a (somewhat) arbitrary name that is (somewhat) attached to the Wii and use the Nintendo Xii (sounds like how Americans say "Z") ... This will be Nintendo's 12th system (NES, SNES, N64, Gamecube, Wii, Gameboy, Virtual Boy, Gameboy Color, Gameboy Advance, Nintendo DS, 3DS, Xii), it sounds similar to Wii, and X follows W in the Alphabet. One reason I could see Nintendo not taking this approach is they don't want market confusion between the XBox line and Nintendo's products

the wii 2, or the daulphin

Just Nintendo ... heard it somewhere and found it nice and simple.

Uii

The Nintendo.

Nintendo 7.

It worked for Windows, why not for Nintendo, too?
It's their seventh console if you count the Virtual Boy as console.

Of course you can use all the controllers.  That would give you the N64, GC, Wiimote, Wiimote/nunchuck, CCP, Wii U controllers.  At least 6 different control options.  I am however warming up to the Ultra Dolphin Revolution Cafe as the name though.  That would put the name in the correct order of release.