Always interesting these types of discussions.
You should absolutely, every single time you buy anything, buy on condition and condition alone.
I would rather buy a well looked after car with 100k on, than an ex rental car with 30k on.
So, consumables (tyres, discs and pads, service items, cambelt (if it has one), exhaust and clutch), are they in good condition and any bills to show they’ve been replaced recently or in the case of the exhaust, does it look good with no obvious damage or bad rust?
Check all the fluid levels and if they’re bang on or pretty much bang on and there are bills to support the consumables being changed and the engine sounds ok and the car drives well, then you should be good. In terms of miles, if you’re looking at 2 absolutely identical cars but one has 70k and the other has 108k, then there should be a worthwhile difference in the asking price to the tune of 4/500 quid.
Always buy a car with a spare key, because if you lose a key, it can be massively expensive on newer cars to have this resolved. If it only has one key, then bargain hard, specially on a mini, as the world is not short of mini’s for sale!
Please keep in mind that your daughter is going to bash it off kerbs, posts, barriers and potentially other cars and spill drinks, food and makeup all over the inside, so a first car should never be a precious thing.
Buy her a car keeping the above in mind, get base AA cover and then add their mechanical breakdown cover for an additional £50 and that will cover the car for mechanical failure as the result of a breakdown to the tune of £500 and you can have 5 claims per year with a small excess each time, so up to £2500 in a year (I have this cover on the 3 cars we have and used it recently when the starer motor went on my BMW 330I sport. I called them, they came and took the car away, fixed it the next day and then collected me and took me to it and all I had to pay was a £35 excess for the claim.)
So, condition, condition, condition. AA mechanical cover and let her rip!