Posted 07 May 2012 - 11:19 AM
The coil of the solenoid, which would have been prime suspect, is not failing, as it clicks. The only other thing to fail is the contacts. I suggest taking one apart (probably needs a lot of brute force) to see if the contacts have been badly burned.
Dan made a good point. A simple test for that is to put the headlights on, and try to start. If, when the solenoid clicks, the lights go out, it is definitely the battery or its terminals. If they just dim slightly, which you may not notice, the battery and its terminals are ok, but the engine earth may not be. If they dim a really long way but don't go out, the starter may be jammed.
In any case, if your battery terminals are the pot type, change them for the clamp type, as the pots always fail sooner or later, and are the cause of much misery.