One thing you have to remember about voltage is that it's always relative. One cannot speak of an absolute voltage; one always has to talk about the voltage between one point and another. This is in fact where the notion of "ground" in a circuit comes from: it's the node (a node is a part of the circuit that is all connected by just wires) that one measures all the voltages relative to. The term further comes from the notion of "earth ground" where one measures voltages relative to the local surface of the earth.
Static charges are charges that accumulate relative to the earth and can reach in to the tens or even hundreds of thousands of volts. Voltage basically means how eager electricity is to get back to the point the voltage is measured from. Since we're talking about a huge voltage between the ESD source (in this case, your body) and the earth, the electric charge wants to find any path it possibly can to the earth. If this means going through a bunch of sensitive electronics, it will do so. You want to make sure that a path exists that bypasses all your electronics. This generally means tying all the exposed metal bits to a something that is as directly connected to the local earth ground (that third prong on an electrical outlet or a ground stake) as possible.
I usually use green AWG16 for grounds in my arcade cabinets, but that's because I picked up 1000ft of the stuff for cheap a while back, and it's the right (generally accepted) color. Ground braids work well, too, and AWG16 is probably overkill in many cases. Bigger is better as it represents a better path for the electric charge to take (and it will take the "easiest" or best path it can find), but even a relatively small wire like AWG22 represents a better path from the electricity's point of view than a complicated transistor circuit like the processor on your encoder board.
The gist of this is that, yes, AWG22 is probably fine. By convention, it would be green (ideally with a yellow stripe) rather than black, but of course the electricity doesn't care what color the insulation is, so as long as you know what everything does, it's OK. It is very important that earth ground actually make it into the scenario, though. You should absolutely ensure that your outlet is properly grounded. Many devices that have 3 prong plugs represent a mild shock hazard during normal operation (and an extreme shock hazard if an internal fault occurs) if not plugged into a properly grounded outlet.
Again, if you have three-prong outlets that aren't actually grounded (or that have AC neutral tied to the ground - have an electrician inspect this if you're unsure), you really should have that resolved as it is a pretty major safety and even fire concern.
Ideally there is one and only one connection between earth ground and your DC power common ("ground" aka the big black wire) inside your arcade cabinet. This is usually placed near the power supply, and PC supplies all do so internally. From there, everything is separated into earth ground and power ground to prevent interference (it shows up especially badly and obviously on monitors but can cause all sorts of other difficult to debug problems). Earth ground is only used for ESD and safety management (i.e. tying all the non-electrical bits of metal together) while power ground is used for all the power and signal wiring. None of this separation is strictly necessary, but some electrical codes and standards require it as it also has some subtle safety implications.
Copper is in general very expensive right now, so just about any source of copper wire or braid will be expensive. I've bought most of my wire by deal hunting on eBay and local surplus shops (mostly eBay).