Without someone having looked at the device or the specs there is little point in speculating.
If it is the same connector, and connector design is a pain in the arse so if they have one that works and has been proven reliable (I don't know if I have ever heard of one going, however unlike charger ports it is not as likely to be brought up or cross my bench when the remotes were otherwise so cheap) then it is certainly a reasonable thing to see, then there is a breakout board, a bit simplistic if you fancied a simple adapter or extension socket like we have seen for other consoles for years but it will get the signals out
https://www.sparkfun.com/products/9281
The wii was seen to use i2c which is one of the more common chip to chip communication methods. To that end you can probably find some means to convert if it is that, however just because it is used there does not mean it is used here (plenty of four wire communication protocols available after all). If you wanted to buy such things in ahead of time and maybe rig up some ways of reading i2c (available on some scopes, probably a bunch of usb devices and the classic thing we see in these circles is the bus pirate
http://dangerousprototypes.com/docs/Bus_Pirate ) and whatever else then you could probably sort it out yourself with relative ease -- it is likely not a massively complex two way logic but simple button presses. Bonus points is you might even be able to make a way for you to have rapid fire, speedrun style button combos and the like for the nes mini.
Even without all that you could head down the following path
http://www.instructables.com/id/SNES-Teensy-USBPS3-Gamepad/ , such things work for basically any controller when all is said and done. It does not necessarily involve cannibalising the controller but if you are not going to do that and try to have it all integrated it is harder.
That is the basic electronics answer, others have given the more practical ones though.