From a pure hardware & firmware perspective, this is what I would ensure the Wii U's successor had:
1. The UI needs to be fast - This is a real sticking point with me on the Wii U, given that it's a console allegedly "more powerful" than the PS3 or 360's, yet its UI is painfully slow to switch between Apps or settings. It's so abysmally tedious to navigate that it's actually a large part of why I don't use the console. This issue was made laughably more obvious after I got my PS4, whose UI has problems of its own but it's ridiculously fast to navigate and switch between tasks on the fly. Whatever the Wii U's successor is needs to have a ludicrous amount of RAM so it can have similar capabilities. Naturally, that RAM increase would benefit games as well.
2. The controller needs to be standard - The Wii was a fluke. People don't generally buy consoles for their controllers. They buy them for the games. The GamePad is an albatross around the Wii U's neck, keeping Nintendo from dropping the price while simultaneously banking on a feature that can be and has been replicated on other devices. And yeah, I've never cared for it as a controller, either. A nice refinement on the Pro Controller would work just fine on the next console. Gimmicky hardware B.S. for the sake of being gimmicky is part of what made the Wii U the failure it is and drove developers away.
3. Achievement/Trophy System - I get that many of you either don't care or actively dislike achievements/trophies. That's fine (and I understand why you feel that way), but you can't deny that such systems are popular on other consoles and it's not like Nintendo hasn't dabbled in such things before with games like Xenoblade. Implementing an achievement system in with something like Miiverse would really help foster that feeling of community on the next console.
4. Technological Parity - If Nintendo insists on competing in a console market that includes high-end PCs; the PS4; and the Xbone, Nintendo's next console needs to have technological parity with their competition. I understand the arguments people have for why to not do so ("driving up the cost of games", etc.), but the greater the technological divide between the consoles, the less chance there is that 3rd parties will develop cross-platform ports. Even the GameCube, as poorly as it sold, got cross-platform 3rd party ports because it was technologically in line with the other 2 platforms. Going with 2005 tech didn't make the Wii U a hit OR a profitable failure like the GameCube, and we've all seen what its 3rd party support looks like. Nintendo needs to swallow their pride and join the rest of the industry with their next console.
5. Social Network Integration - This is one of those things that I honestly don't give a **** about in my console gaming experience, but whether I like it or not people seem to be eating that stuff up on the PS4 and Xbone right now. Miiverse is a nice step in that direction, but Miiverse is limited to Nintendo's own hardware and a website that no one uses. Meanwhile, I can play a PS4 game, snap a picture, and post it to Twitter in seconds (with a snarky quip) where people actually do see it. No one knows what the next big Social Media network will be since they change all the time, but whatever Nintendo's next console is needs to support it.
Just a few things there. I'm split on whether or not a console/handheld combined device is a good idea, mainly because a device powerful enough to be a modern home console would have **** battery life as a handheld, and what works very well as a console controller doesn't really work as a handheld controller. By the time Nintendo's next console releases, it will probably have to be a combination device, but I just don't know what that would look like.