Make sure the solenoid is securely grounded. IMO its worth taking it loose tto check for hidden corrosion underneath.
The suggested test is to momentarily jumper to the side terminal on the solenoid, which should be the same as pishing the starter button. I think it unlikely you will see a difference, but its worth a try. If you do, then it points to a wiring problem or bad switch.
Most likely a bum solenoid, but check the other stuff before shelling out for a new one.