In this example we're trying to repair an Atari joystick with a bad 'fire button' but the concept is the same for other problems.
1. Take the joystick upside down in the palm of your hand and Remove the 4 screws on the bottom of the joystick. (see photo)
2. Pull off the bottom joystick assembly to separate it from the upper assembly and put it aside for now. Be careful, there's a little spring under the fire button.
You can use this opportunity to inspect for cracks on the ring of the joystick control assembly that has the 'Direction Tabs'. (see photo)
3. Identify the fire button and carefully pull back the protective film around the fire button that seals the contacts. (see photo)
Ideally you want to just fold back a small section of the film to expose the contact.
4. Remove the upper contact. (see photo)
5. Clean both the contacts on the PCB and the removed upper contact with either contact cleaner or varsol.
6. Let dry (important)
7. Put the upper contact back over the PCB contacts and pull the protective film back over to seal.
If the original film has lost it's adhesiveness and it won't stick when you fold it back over the contact, I usually use a small hobby knife to cut away that portion of the film. I then use packing tape (cut a small section just large enough to cover the contact). You should clean the area around the contact that the new tape will cover (there's often residue from the old film, varsol works well).
8. You can actually hookup to your Atari to test the fire button before reassembling everything.
If cleaning the contacts fails to resolve your issue then you can either replace the PCB assembly or try repairing the circuit traces with something like a 'nickel conductive pen' (available from MG Chemicals : CAT. No. 841-P).
9. When putting everything back together, it's easiest to put the joystick PCB over the controls assemblies first so you can make sure everything is lined up. The fit the lower assembly back over the upper assembly. (see photos)
Everything should fit together nice and flush, don't force anything (the wiring harness can sometimes get in the way. As always with this vintage gear Don't Over-tighten the screws.