That's your minimum for a $200 - $300 handheld device? Prepare yourself to be disappointed ;)

My guesses:

-It will be used in tandem with home console NX as a controller.
-The NX home console will support multiple NX handheld/controllers.
-It will be very affordable.
-It may use Sharp IGZO displays (depending on costs).
-It will have a single screen.
-It will still have optional 3D - They'll likely make 2 versions like 2DS and 3DS from the beginning.
-It will be more powerfull than PsVita but nothing mindblowing.

I'm speaking more from a creative stand point and what the NX (portable) needs to accomplish to fulfill Nintendo's needs as a company. 

I think it's fairly plainly obvious that Nintendo as is cannot even handle supporting the 3DS and Wii U simulatenous. Therefor having a further hardware generation where both the handheld and console are even *more* powerful yet require distinct software libraries and thinking that will be anything but a trainwreck of a disaster isn't logical. 

That's why I think strongly that the NX portable *must* be able to play all the main Nintendo games. Splatoon 2, Mario Kart 9, the next Smash, the next Zelda, etc. etc. 

And sure I think it's possible to have a very, very, very powerful portable at $220-$230 or so (low enough in price that Nintendo could drop to $199.99 in short order). The Apple A9X which is probably more powerful than Nintendo's Wii U chip is a $40-$45 chip, get yourself a small-ish (6-inch) 1280x720 screen, that should be dirt cheap these days, and you've still got plenty of budget left for everything else provided you're willing to sell near cost. 

Add in the novelty of it being able to stream games to a TV (therefor making it a portable that's also a mini-console). That adds value to the proposition. Throw in a free Amiibo and bundle an Amiibo starter game with the system. That's way better value even at $250 than the current 3DS is at $200 if you ask me, and it's reasonably different from your average budget Android tablet. 

Some people seem that understand that Nintendo will make handheld with $150-200 price point and that hi res display (1080p) requires stronger and more expensive specs in order to run games on that resolution, so 1080p resolution is very unlikely while 480p-720p is very possible.

The next handheld won't be able to run Wii U games on its own. But it will be able to be used as a controller for Wii U games.

I think Nintendo understands the home console and handheld audiences are generally looking for different experiences. Sony and Nintendo tested the water with popular home console genres on their handheld consoles (Sony inparticular) and it didn't do well. Fortunately for Nintendo, they have popular mobile games and that fueld 3DS's success.

3DS was inevitably going to perform less than DS. DS achieved an unprecedented level of success just before the market was moving to smart phones. Pretty much everybody having access to a smart devices obviously impacted dedicated gaming devices like Vita and 3DS.

Specs don't matter at all at this point. What Nintendo really needs, is new ways to play. Continue to innovate and give people something they can't get on their smartphones. Because otherwise, they will just stick to their phones.

I think Wii U (and the entire Wii brand) will be quickly buried by Nintendo. Retailers are not going to want to stock it any longer than they need to either, it'll be phased out quick and Nintendo will not want to taint the NX brand (whatever they decide to call it) by associating it with the failed Wii U. So probably very quickly it will be gone from sight. 

It would also be a marketing nightmare for Nintendo as it would start the whole "well what is this thing? Oh it's a controller for that Wii U thing ... wait what, this is supposed to be a new system? I'm confused" type thing. Maybe a company like Apple could navigate that, but it's well beyond Nintendo's limited marketing abilities. NX has to be a clean slate for that reason I think. 

Just like you almost never heard a peep about the GameCube once the Wii was fully unveiled. 

Yeah, I can see that happen.

They ditch the Wii U tablet controller for the NX console and the WiiU BC is limited to games that work with the Pro controller.

A combination of NX console and NX handheld (as controller) gives you full WiiU BC.

That's impossible.

To the OP, he says a lot of things that are obvious. Of course the next handheld will be more powerfull. Before the mobile boom, the natural progression would lead to specs around ps360. Since the mobile growth increased demand for portable specs, the 3ds successor should be somewhat more powerfull than a ps3 or x360. 250-300 gflops gpu, 2 gb of ram, better cpu and access to the most up to date open gl, shader model... All for $199 at launch. Just forget a 1080p screen, 720 is more reasonable. I also think the 3d is cheap (and stable) enough that they will keep it forever.