Swap it with a different one. Failure is very rare. Based on your tests and observations, I'd suspect the neutral safety switch.
When you can catch it not cranking, grab a test light, ground it to the engine or battery negative, then test the four terminals in the starter relay socket. (Middle terminal is not used). One will have battery voltage all the time. One will have 12 volts when you turn the ignition switch to "crank".
Next, clip the test light lead to the battery positive post and probe the two remaining terminals in the socket. The test light will light up all the time in one of them because it is finding low resistance to ground through the starter solenoid coils. The other one will read to ground through the neutral safety switch when the transmission is in park or neutral. If you find ground in only one terminal, suspect the neutral safety switch. The starter solenoid windings hardly ever go open circuit and you already found it working when you jumped the relay terminals.
Saturday, December 11th, 2010 AT 5:35 AM