All true, but it depends what *kind* of click...
The Gryge deduces:
He says he's getting a loud 'click' . . . . which would indicate the solenoid and it's wiring are working. Dead solenoids don't click and dirty solenoid contacts (rare) are generally overcome with a few clicks.
No, *electrically* dead solenoids don't click, that's right... ("electrically dead" as in, the thin wire that the coil consists of is broke.) But, solenoids -- at least on older Volvos -- can get "stuck" in a purely mechanical way, that doesn't mean that the solenoid itself is "dead".
I think this might be a good place for short explanation of what a solenoid *is*, for those -- like Thane, I suspect -- who aren't quite sure... So: A solenoid, at least as the word is used in automotive parlance (dunno if it's used in other ares too?), is a dual-function device based on an electromagnetic coil. One of its functions is to close an electrical circuit that provides the starter motor with power; what is generally known as a relay. The other, I'd say the solenoid's "main" function, is to mechanically connect the starter motor to the engine. This is done by the simple expedient of a lever which, when pulled forward by the solenoid's electromagnetic coil at its upper end, pushes a little cogwheel that slides on the axle of the starter motor backwards, to mesh with a rim of cogs on the outer edge of the flywheel and thus allowing the starter motor to turn the engine's crankshaft. When you release the ignition/starter key, a spring pulls the little cogwheel back(*) to its original position, thus decoupling the starter motor from the engine; you don't want the starter motor to turn with the engine all the time.
What can go wrong here, in a non-electrical sense that Andrew apparently didn't consider, is that sometimes the cogs won't mesh -- despite being shaped so as to facilitate meshing -- but just slam into each other. This still makes a click, but not the satisfying full kla-chink of a little cogwheel sliding all the way home and gears meshing; it is, rather, a shorter sound, a dull, metallic "clunk" like letting a hammer drop just half an inch onto an anvil that's almost buried in earth.
This doesn't happen all that often, and I don't know if it happens on your or other cars at all; maybe it's limited to just the old Volvos I'm familiar with... But if this is your problem, Thane, then the fix is relatively easy: Turn the engine by other means. A few degrees is all you need; just so the cogs on the outer rim of the flywheel move a fraction of an inch. If you have a manual gearbox, put it in high gear and give the car a little shove; otherwise, try pulling on the alternator-(formerly fan-)belt.
The rest of Andrew's advice, on battery terminals and wiring and so on, I agree fully with.
(*): i.e, on a longitudinally mounted engine, forwards.
