I have the same problem on my '88.  There is a high resistance contact somewhere in the starting circuit.  The clicking you hear is the cold start fuel injector which is a good sound to have.
I fixed mine by wiring a momentary contact startwer switch under the radio.  I brought a hotwire from the battery with a fuse to the switch, and from the switch to the starter.  But I put in a parallel wire from the regular ignition switch.  So it is a "Y" at the end.  I put a female end onto the starter with an extra wire and male end to flip back to the regular key switch. 

Now if it fails to start with the ket I reach over and push the button to roll the starter manually.  Just a remote starter box.  has the advantage of activating the cold fuel injector for me with out extra work.   

before this I put in a new battery, cables, starter, and alternator.  Wasted a bunch of bucks changing stuff that was just fine.  Still do not know where the real problem is but I know where it is not.  LOL