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