The CVT (auto) gearbox has a well deserved reputation for failure and is very expensive to replace.
There are parts and rebuild kits available, but not all transmission shops are willing to work on them.
With a lot of work and extra parts (including ecu reprogramming to manual), you can swap it to a 6-speed Getrag,
but you need to know someone who knows how to do all that, and it probably costs even more than replacing the CVT.
Changing the gearbox oil regularly may help.
The manual gearbox in 2002 (Midlands) is also prone to failure, but not as often as the auto CVT gearbox.

Power steering pumps do fail occasionally, but not that much more often than similar components on
other 12 year old cars.
Try to make sure the low speed resistor for the radiator fan is working, as it is linked with the power steering
cooling fan,
and high temps under the hood are one thing that can cook the power steering pump.
Make sure the PS cooling fan is unobstructed and working, also.
Also, don't let the fluid (Pentosin CH11S only) run low - the power steering hoses often seep a little after many years.

 

I have an 03/2002 R50 5 speed manual, second hand with 160000km now at 176000km in 3 months. Replaced the PS pump Aus $1000.00, Engine mounts (1 was about to fail so I replaced 3 of the 4) Parts Aus $350.00 approx., Replaced the OLD style fuel filter (you have to replace the whole unit) Aus $400.00 and a new fuel pump Aus $375.00 ! Am I concerned ? no, its a 12 yr old car with poor prior maintenance. Previous owner used the wrong PS fluid (will damage the pump, never replaced a seized PS cooling fan (lazy), failed engine mount, normal wear and tear. Fuel pump and filter, luck of the draw ! At the present time, its running fine. Gearbox is Midlands, shifts without issue and cant fault it but I also don't flog the car to death and don't ride the clutch. Labour on repairs is done by myself so that cuts down the cost. Looking at dropping the sub-frame in the future to replace ball joints and bushing, more a preventative maintenance than current need.

For any second hand car you should expect problems without an exact maintenance record (I'm recording mine) and as its a BMW, you expect to pay for the brand name so if your capable of doing your own labour you can save money. One thing to consider with issues like gearbox failure, forums will only ever tell you about ppls issues when things go wrong, what are the actual statistics of all the Midlands gearboxes that have failed world wide and how were they driven ? how many never had an issue ? A rebuilt Midlands gearbox on EBay Australia in Sydney is offered and almost Aus $3000.00 supplied and not fitted so its not a cheap gearbox. Remember its a 12 yr old car, thing will wear and need to be replaced. Preventative maintenance is your best bet, regular oil changes and filters etc. Don't drive it like you stole it but do drive it spiritedly. Look after it and enjoy it and it will probably serve you well. There are some good websites and Youtube Videos for maintenance etc, Mod Mini on Youtube has some very informative videos and a good website is BillsWebSpace and Pelican also have some good repair/maintenance articles.
With some decent tools, they are not that hard to work on and maintain, also recommend the Bentley manual for ref and repairs.

 

FWIW, my 2003 Midlands just failed at 79,000 miiles, and I just replaced it with a 6-speed Getrag.
I'm the original owner and I treated it well. I realize not everyone's Midland will fail, and in fact, most won't.
The CVT failure rate seems to suggest the question is more when it will fail rather than if it will fail, however.

 

R50 Midlands gearbox

The Midlands gearbox has a magnet on the filler plug as well as the drain plug. BMW say the oil is good for the life of the car but if you inspect the filler plug it may prompt you to change the oil. My 2002 Cooper is my club car and has been hammered on various race circuits and hill climbs for almost 10 years. I change the gearbox oil every 2 years as an investment.

 

The Midlands boxes are really only good for ~2 years per box. Nasty piece of work. I blew up 4 of them Expect a conversion to 6 speed to cost ~$10k to swap over but it's the best mod you can do! Beautiful gearboxes.

Fan resistors only seemed to be a problem on the facelift (06/2004 onwards) models.

Power steering fans and pumps only seemed to be an issue for the original units from the factory. Later cars and replacement units seem pretty good. If it's been replaced chances are you'll be fine.

 

I replaced a fan resistor (actually replaced the whole fan assembly as the fan motor itself showed signs of
wear and reduced performance in addition to the blown resistor) on my 2/2003 R50 several years ago.

 

I stand corrected First one I've heard of in the pre-facelift ones.

 

Putting in a 6 speed box from a Cooper S sounds like a lot of work to save a vehicle with today quite a limited value, buy a facelift car with the 5 speed Getrag makes more sense to me.
Moving on, anyone had any issues with the 1.4 Toyota diesel engine in the Mini One D r50, they seem capable of big miles?