The problem of mini-games IMHO is that they are mini, but you have to have fun doing them dozens to hundreds of times in a game.
I can think of a few possible variants, and I like none of them:
1) puzzle games that challenge your brain, but what is hard for one user may be trivial to someone else. And most puzzle games that are solvable in the matter of seconds get trivial very fast
2) reaction based games. The same problem, very hard for older players, trivial for a 16 year old fortnight player. And personally I really really hate stuff like that. If you really like to play a reaction game hundreds of times in a game, get yourself a C64 emulator and play stuff like Olympic Games.
3) Shooting stuff. Similar to reaction games, also we have enough shooting built into the game, we don't need a second-rate version in a mini-game as well.
4) puzzle game with heavy randomness, for example a card game like poker. May be the most fair solution, but again difficult to get something engaging when it has to be over in 10 seconds. After the 20th time you will have your strategy fixed and mechanically do the same things again and again until you get lucky.