Sounds dumb, but I installed a starter once that would only click. There were two small terminals on the solenoid, and I hooked to the wrong terminal. Not sure what the other one is supposed to do, but I switched and it was good to go. Do you have two terminals? Really I have no idea but it might be worth a check. My problem was on an F-700 with a Cummins.
1991 F-150 XLT Lariat, 5.0L, 5-speed, 120k miles
2015 Honda CRF250L
2014 Escape SE 1.6L
2009 Focus SE
Fords at work: '11 SD 6.7L, '08 SD 6.4L, '06 SD 6.0L, '05 SD 5.4L, '99 Ranger 4.0L, '96 F-150 4.9L, '96 Ranger 3.0L, '95 F-350 5.8L, '94 F-150 4.9L, '93 F-700 5.9L Cummins, '91 F-350 7.5L, '77 F-150 5.8L