In the ballpark of what the Vita already has, much like how the 3DS resembles the PSP.

After the 3DS launch Nintendo knows they need their handheld in the $150-$200 price range, with that in mind I'm gonna say slightly above the Vita.

I'm betting they'll want to get on with the rebranding, and both their next-gen platforms will utilize dual screens.

I say as powerful as PS3. qFHD top screen, with 3D and 15:9 aspect ratio. qFHD bottom screen, with capacitive touch support (it's cheap nowadays) and 4:3 aspect ratio to accommodate buttons.

ITS OVER 9000!!!!!!!!!!!!!!!!!!!!

I hope you mean qHD, as in Quarter HD (960x540) and not Quad FHD as in 3840x2160. ;)

Damn it, got my acronyms jumbled.

Reason I went along with this is because I seriously doubt Nintendo would be in the right minds to just go for VGA next gen.

It will be powerful enough to port Wii U games/port Wii U engines to IMO.

This way Nintendo's dev teams don't have to make a big adjustment in development changes and Nintendo can port games like Xenoblade, Bayo 2, Captain Toad, DKC: Tropical Freeze, etc. etc. that a lot of people aren't going to get to play this gen because they refuse to buy a Wii U (don't think for a second Nintendo is just going to be OK with these higher budget HD games selling sub-1 million, sometimes sub-500k numbers). 

So quite powerful. You'll be able to play the same games at home (Fusion concept), just at 1080p + 60 fps and maybe some other added graphical effects.

The Tegra K1 is already supposedly as powerful as a PS3/360, even if Nvidia is blowing some smoke, by 2015, the K1 successor will match/exceed the PS3/360 level graphics, assuming Nintendo is not launching until 2016, that's another year for that level tech to scale down comfortably in cost for a $200-ish tablet like device with a 6.5-7 inch screen that isn't super high resolution (this is where Nintendo will save some money by going with a cheaper 1280x720 screen IMO).

Going by past Nintendo handhelds and the progression in their specs, a 3DS successor will be almost as powerful as Wii U. But that also depends on its release date.

If it launches in 2016/2017, Nintendo would probably make it as powerful as a Vita or a bit more powerful as they usually make very weak hardware for the time of the release of their handhelds. If it launches later, it will definitely be about as powerful as the Wii U.

It would be more expensive to do hardware weaker than Vita in 2017 or so. I'm expecting an equivalent to a modern budget phone SOC of that time. Something like Snapdragon 400 today. 

Um, Snapdragon 400 uses Adreno 305/306...which is around Vita.

You sure you were not thinking of Snapdragon 805 and Adreno 420 which is some 5-6x Vita?