I used to own a 52 Plate Mini Cooper-S as well as run a small club. Some things in my list will have already been mentioned above.

1) Early gearboxes issues
2) Exhaust hangers rusting
3) Hydro-electric steering issues
4) Coolant header bottle splitting
5) Clutches take some hours to change
6) Electric window issues
7) Central locking issues
8) Lower subframe bushings
9) Fan sticking on

Through my 5 years - 70k miles ownership I had the fan stick on, lower subframe bushing replaced, electric window and central locking issues.

On the plus side, the difference between a One and a Cooper (petrol) is a remap so you can obtain some gains with something off the shelf.