I thought I'd try to speculate about what Nintendo might do, based on their past and present.
Price range: $250-300. The Wii U proved that $350 is too much for the general public to swallow. The Wii was about $250 and practically flew off of the shelves. Any lower would incur too many compromises for the console to be enjoyable.
Performance: Improved but not shockingly. At the bare minimum I can see Nintendo taking what they learned from developing for the Wii U and addressing any bottlenecks they have found. My assumption is that the possible maximum would be just enough that, for example, Mario Kart 9(10?) is 1080p with six racers (two on gamepads)
Optional 2.5" internal hard drive: Most consumers would stick with disks and so won't need a lot of internal memory, but there will be a lot of people who will have a large digital collection and so need more than simply 32gb or even 64gb of flash memory. My guess is that there will be a pro bundle with a 500GB drive preinstalled, but there shouldn't be anything stopping you from fitting a drive of your own into the budget model.
Game media: Most likely the exact same format as for the Wii U, maybe with support for dual-layer disks for the few (third-party) games that need more than 25GB. Very few Wii U games exceed 10GB, and it's not very likely that the Wii 3 would drastically increase game size.
That's all so far that I thought of that I feel certain of. What do you think Nintendo will do?