Jim,

If the starter was trying to engage, I believe that there would be more than a "click"... I'm thinking more "clunk" or a grinding noise. Probably not the starter. I would lean toward the solenoid.

Also (I'm trying to confirm this) I think that some solenoids, or more likely the ignition switch/relay, will disconnect normal electrical circuits during a "start" so that all current is used for powering the starter... ever notice in some cars how the lights will go out just before the starter is engaged? Anyway, there is a remote possibility that the starter switch or even the neutral safety switch is involved in your problem.

Maybe this will get someone else to remember a similar problem.

Dan J