Classic old Mini problem, battery in the boot, starter in the front, crappy connections accross the subframes and body, as Tom says try to take direct connection from battery earth to Engine block, needs to be chunky jump lead type cable, first thing I always used to do in my yoof, run direct eath cable to Block on minis.
Re temp, its all additive, battery gets cold overnight, cables colder, engine colder, it might not be earthing but for sure its the first thing to verify.
Could try a quick visual, look for a braided flat copper strap somewhere around the engine steady, going from Block to Bulkhead, quite often go greenish, corroded or just plain snap, if you don't want to run a cable all the way from battery you can try replacing the braid with a couple of new ones from block to bulkhead (running from different places) one of the dangers if the earths are dodgy is everything can earth back through accelerator cable and things start getting hot .
Tim
Edited by - tbird on 22 Jan 2013 23:04:15