Last week my 5.4 F150, just clicked once when turning ignition on. I replaced the starter thinking that was the issue. Same thing. So I crossed the new starter solenoid, couple of cranks started right up. Got the truck back home and put the old starter back in. Worked fine for 5 days, now it is doing the same thing. When shifting while turning the ignition it clicks when hitting neutral and park. I replaced the positive lead on battery and cleaned grounds. Still broke...

 

OK, now it doesn't even start with the solenoid jumped (plenty of sparking going on at the firewall and starter). Replaced the ignition control module, leaving it in I guess. OH yeah when turning the key over it will lock and unlock the doors randomly if that helps.

 

Slick, ya gotta be slicker than the fault.
....If it won't turn over with a starter bypass and your sure the cable connections and grounds are clean and tight, a possibility is the battery has an open intercell connector.
....A hint to this is observe interior lights when you try to crank.
....If they are on as near normal then go out when you attempt to crank, the battery path has a near open but just good enough to power a few amps of draw until the starter needs a hundred amps or more then there is nothing left for the interior lamps and other electronics to work on due to the large voltage drop accross the high resistance open condition.
....Another way to see this is with a voltmeter in the cigar lighter socket to monitor the voltage at cranking attempt.
....If it goes way down to near nothing, the above issue is a possibility.
Good luck.

 

I will take a look at that. Too dark to mess with it now.