Hi Tom,
before taking too much apart you can do a few simple tests to pinpoint the problem first.
The most likely culprit if your battery and connections are all good -including the ground strap connections- is the cutoff switch in the trunk. Try connecting a booster cable between the ground battery terminal (normally the rearmost one) and a clean ground on the chassis somewhere, this will bypass the cutoff switch. I'm assuming your car has a functioning switch.
You can isolate the problem a fair bit by turning on the headlights and trying to start.If putting the headlights on means the solenoid no longer clatters then you have a bad connection somewhere. If the headlights stay much the same brightness or dim just a little then you might need to tighten the solenoid mounting screws as it earths through them. If you press the solenoid button with the headlights on and they go very dim or completely off then you almost certainly have a bad connection in the circuit.
A machine-gunning solenoid is unlikely to be caused by a bad solenoid. The sequence is: solenoid pulls in, large current tries to flow to starter, voltage drops below the solenoid operating level, solenoid releases, voltage rises again... This is almost always caused by a high resistance connection in the starter circuit but also could be caused by loose solenoid to body bolts, an internally shorted starter or a battery that isn't performing correctly.
Andy.