Being of a certain age I have spent many hours (some happy - some definitely not) working on classics i.e. original Minis, and especially the Cooper S. From an electrical standpoint they were never 100% reliable and from memory the main culprits (assuming this is an early generation Mini) were cracked distributor caps (I always carried a spare), the coil and the rotor. There were also occassional problems with earthing routes.
Although I haven't played with one for some time I also remember on one of my Cooper S I installed an electronic ignition unit which was a lot more reliable. I would imagine these are probably still available.
As a matter of interest the last Cooper S I owned produced just about the same amount of BHP as the new Cooper... A fraction under 130bhp courtesy of a highly modified engine with twin Weber 40DCOEs..... !!! It used to frighten me sh*tless