Tap the starter with a hammer. If its the solinod then I would think a quick tap would make it start.
Also
Maybe check that your S termnal on the starter is getting 12 volts when the key is turned. I think you can get a starter button at Sears. Be very carfull when doing this.
I have seen this with corroded connection at the battery. Try twisting the connection and double check that the connections are tight at the batter/starter and the also the ground. I had ground that was not tight and gave me fits like this.
