I've had 2 - a 2001my and a facelift 2007my
The 2001 common problems were -
Gearbox - Midlands gearboxes notorious for going wrong so budget for a recon as a precaution
Airbag warning lights - sliding front seats stretches the cable underneath causing the warning light to come on, BMW can supply an extended harness
Waterleaks - high level brake light leaks (remove, seal and refit), rear light units leak (remove, seal, refit), sunroofs leak
Knocking from rear over bumps - replace rear ARBs (£9 a pair from BMW - 1/2 hour to fit)
I replaced quite a bit of trim etc on both cars (i like my cars immaculate) and found genuine BMW parts to be quite reasonable
Timing chain problems were on the later Cooper S models (2007>) with the peugeot engines - the chains stretch, also common on 1 and 3 series BMWs