My Atari 2600 has a broken knob

The switches on the 2600 tend to become loose over time. There are generally three things that go wrong with them.

1) the solder connections work loose. If you open it up and look at the solder points, then loosely wiggle the switch. If they are loose, re-soldering is a simple matter. Typically there's so much on there that simply melting the existing solder and allowing it to re-solidify is enough to repair it. Make sure you don't let two solder points merge together.

2) the switch loosens and falls apart. The switch itself is held together by a metal housing with four prongs wrapped around a plastic base. These prongs are notorious for not being bent enough and over time work loose. Carefully tightening them down with a pair of pliers will help. Do not squeeze too hard or you will do irreparable damage to the plastic, and/or make the switch too hard to move.

3) The leads inside the switch become dirty. You must then un-bend the above mentioned pins, remove the metal casing and lift it away. This will essentially allow the switch to fall apart, so note how it's put together as you remove it. If the connection pads on the inside are dirty or black, carefully scrape the metal contacts with sandpaper until they are shiny silver again. Reassemble the switch and tighten the prongs as above.

Don't be afraid, these things are built like tanks. In fact, these are good switches to get practice with fixing before you move to more modern switches.