my 86 f150 6 cylinder sometimes will not start. Key turns but nothing, no click, no chug, just nothig. Has plenty of power all lights work. Other times it starts right up.

New set of ignition wires...couldn't hurt~!

May not be the starter solenoid but the brushes in the starter are bad.

ignition coil module?

of course you got a Brand NEW battery and new cables too~ be sure the cable to the block also can make 50 amps~

Try tapping on the starter with a hammer while trying to start vehicle at the same time if vehicle starts replace starter.