Mitsubishi Repair: starting problems, starter problem, starter relay

Question
mitsubichi montero 1990 automatic
 It will not start after sitting out in cold weather. The battery is fine....the lights come on, the radio works etc.  All it does is make a click and will not turn over.  It has a relatively new starter.  All the wires, cables etc. have been checked out and tightened... it will not start when jumped and then it will just randomly start.  It has a newer engine, fuel pump and all new belts. Has several mechanics baffled.  

Answer
Sharon,
Though you mentioned your starter is relatively new, it still appears that it may be a starter problem; or perhaps your stater motor was installed improperly and misaligned.  This can cause the starter to stick.  

I will assume your mechanic actually load tested your battery to verify it was fine.  Your lights and radio can function properly even with a weak battery because they draw relatively low amounts of current.  But on cold mornings, your battery needs to use all its power to turn the starter motor.

The click you hear after you turn the key most likely is the starter relay.  It's possible this relay or the connections on the relay are weak and needs to be replaced.  Hopefully your problem will be something simple, but you may need to have the starter replaced again.
Good luck!