It's important to check that the engine isn't smoking from the exhaust - that means that the piston rings or valve stem seals need replacing, which is a very expensive job. Best way to check is to let the car warm up, leave it idling for 3 – 5 minutes, then rev the engine (as if you were pulling away gradually from traffic lights). If you see smoke coming out of the exhaust, walk away.
This issue is mainly on older Cooper S with the Renault engine, you shouldn’t see it on the newer cars with BMW engines. Definitely ask the owner if the car uses oil – he/she will probably say yes, it’s part of the design of the engine, but then ask them if it uses excessive oil (then carry out the test above). Also ask the owner if the car has any ‘known‘ faults – is the battery OK. The battery needs to be coded when it is replaced, so if it’s had a new battery recently, that’s definitely a good thing.
Also, check that all doors open/unlock correctly – if not, that is another issue which means a difficult replacement of the door lock mechanisms.
Check if there are any squeaky noises coming from the suspension – if yes, that means bushes need to be replaced.
A lot depends on the age of the car, but anything approaching 100k should have had the timing chain and tensioner replaced
And of course, it’s very important for a Cooper S to have had regular services – don’t buy one unless the owner has proof (receipts, service book). Oil changes should have been regularly carried out, with Mini’s, it’s actually important to change the oil more frequently than the recommendations. If any warning lights appear during the inspection/test drive, walk away.