The little green wire up by the switch is just to make sure that all the metal faces of the switch (which switches AC) are safely grounded. In most PC cases, that would be a redundant (but still welcome!) feature, as the supply's enclosure is ground, and the enclosure is mounted in the metal box. However, some PC cases have plastic fronts with no connection to the rest of the case, so it's not always a redundant thing as in this case it serves to ground any front metal that may be present but unconnected to the rest of the enclosure.
FYI, in PCs, ground is ground is ground is ground. That is, earth ground (green wire in the AC plug) is frame/chassis ground is power ground. They're all tied together inside the power supply. However, it's still a very good idea (and it is always done) to run dedicated ground/return wires for all your power. This avoids currents on the chassis which cause EMI and ground bounce. In many cases, the metal case is required to meet FCC Part 15B which is yet another reason it is "ground"ed.
This commonalty of grounds is not always present in other situations. There are some disadvantages to doing it that can lead to things like "hum" on your audio due to multiple ground return paths. A/V equipment specifically avoids doing this common frame/earth/power ground thing for reasons such as this. The setup in arcade cabinets varies.
As for what to do with the little green pigtail, I just take a screw and screw it directly to the metal face on the switch when the switch is "loose" or mounted to something nonconductive. This gets it out of the way and fulfills its safety purpose.