I would have made the CPU quadcore instead of triplecore (same chip due to backwards compatibility), a somewhat better graphics chip (Probably 512 VLIW4 cores instead of 320, GCN weren't ready for such a development just yet back then) and included a standard internet connection socket (instead of needing a special USB Adapter for this). Oh, and maybe produced the chips in 32 or 28nm instead of 45 to lower the heat output and keep the system as small as it is. And 4 instead 2 GiB RAM.
On the software side, I would have lauchned the console with a bit less games (there where 23 at launch!) and spread their releases a bit out to fill out the gaps a bit until spring. Nintendo was unprepared for HD Graphics, but that was one of the 2 reasons why 3rd party came first (the other being them complaining not selling enough games on Nintendos consoles due to the first pary games) and could have worked out for both Nintendo and the 3rd party developers, but sadly it didn't. It was a nice try from nintendo, but it not only failed, but backfired horribly, someting one couldn't know for sure during development. It was risky, but Nintendo isn't exactly risk-averse when it comes to trying out new ways or new things. i would also have tried to attract more Indie devs and bringing out Indie (and possibly Virtual console) collections on disc to fill some later gaps in the software lineup.
As for the name, I would have called it Wii too (for those who don't fully get it, a double wordplay on we, too, aiming at the core gamers, and Wii 2)
Prices would be about the same as they where.
And no, not cutting the Gamepad away, instead opting for the possibility to connect more than one of them on the console (which was actually considered by Nintendo, but not feasable because the console was too weak for this). I would however have reversed the button/Thumbstick layout on the right side to match the one on the gamecube controller, and maybe opted for a stronger battery if this was possible ont he budget back then.







