The reason why early BMW Minis, the R56s, are cheap is because they are absolutely bloody AWFUL. Up until 2011 the petrols used a Peugeot 'Prince' engine which just doesn't work in the Mini.
I know because I have one (and I really wish I didn't). I bought a 'really good' low mileage 2010 Cooper for my daughter last year. It cost £2,800. It has done around 500 miles since then and it now owes me £4,500 and I have done most of the work it has needed myself!
There is a list as long as your arm of 'inherent' or 'common' issues with the R56s, from the infamous 'death rattle' to melting thermostat housings and everything in between.
Other than that they are not very nice to drive, small inside with weird interiors. They do handle well, though, if they work for long enough!