Original (and several fake) Wii Classic Controllers are compatible with the NES Mini. When you play NES games which only requires A, B, Select & Start buttons, they are mapped automatically on the respective buttons on the Wii Classic Controller. Original controllers can just be plugged it and you're ready to play, but for the fake ones, you need to install the latest Clovercon hmod from hakchi2.
For the mapping, I assume (have not tried it myself since I got a lousy fake one) that the Wii Classic Controller will map GB, GBC, SNES games --in case you installed them on your NES Mini-- as well perfectly, but maybe for other platforms such as Genesis, PSX, etc, you have to check the mapping via RetroArch menu (Select + Start while you're in the game).
Hope this helps.