I would bet that you have a bad ground. By looking at the black positive terminal, it seems you have a high amp draw that is looking for a good ground. With the solenoid clicking rapidly, I would think the battery has enough power and the clicking tells you that the switch in the solenoid is moving.
Check to see if the solenoid is well grounded. Do you have ground wires to both, the frame and also to the engine. It also doesn't hurt to run a ground wire to the dash tower as well.
In your pictures there is a fair amount of corrosion, so I would check all connections. It also looks like the ground cable is only one main cable without any other grounding leads coming from it to go to the engine,frame or tower.
On a second thought about the battery having enough power...... it may need to be charged after seeing a high amp draw like it seems to have seen.