If the battery and starter check out good and you are only getting a single click, the problem lies in the starter solenoid or relay. Not sure where the relay might be but the solenoid is usually where the positive battery from the starter connects to. There are at least two wires at the solenoid - the battery cable and a thin wire. check the remote starter solenoid usually monted on right side of engine bay
your battery is dead or the starter solenoid
Batteries don't "click," but other parts do. If you're hearing that "click," it's probably the starter solenoid trying to engage. Either the battery's charge is low, the starter solenoid has failed, or the starter motor itself has failed. On some older cars, especially Fords, the "click" could also be a remote starter relay.
A fast rapid click is an indiction of a weak battery. A single click is an indication of a failed starter.
If it is a single hard click each time you turn the key the starter is going out and will need repaired/replaced.If it is a single hard click each time you turn the key the starter is going out and will need repaired/replaced.
To give a generic answer, it kinda depends on what type of click. If it is a rapid click-click-click-click... it is probably a dead (or near dead) battery. If it is just one solid "click", it is probably your starter. The single click is the sound of the selinoid clicking in to try to engage the starter. If the starter is faulty, it won't engage and turn your engine over to start.
A single click usually indicates a fault in the starter relay or solenoid - could also be a loose cable or wire.
If a starter relay or solenoid is bad, usually you will only hear a single click when you turn the key to start the engine.
The most tell-tale sign is a single click when you try to start the engine.
If there is a single click from the starter area, the starter is bad. If not, it could be an ignition switch.
starter soliniod
If it is a single click, the problem is either a loose battery cable or a faulty starter relay/solenoid. If the symptom is rapid clicking, the battery is probaly low.
could be a number of things. did you try giving the starter a little rap with the handle of a screwdriver. sometimes that'll unstick a stuck starter...also could be a bad ground wire, corroded inside the sheathing. Also check relays