Huh--this is an interesting one.
I pulled off the cover plate and the two hot wires are connected to the plug, and there is a very thin ground wire in the box not connected to anything.
If I were to convert this to a 6-30 outlet, is that thin ground wire enough?
Well, that depends on what gauge a "very thin wire" is.
There is a specific table of how thick the ground wire needs to be for how many amps the circuit is rated for. For a 30A circuit, the ground wire needs to be 10 gauge.
Equipment Grounding Conductor Table 250-95
Need to split this up:
Grounds have to go back to the breaker box by code.
I'm not quite sure about that one. I know neutral lines do have to go all the way back to the panel. I thought grounds might be able to tie to grounds in other places. Need someone with a better answer on that one.
I'm not sure what you have with a 2 pronged 240 V circuit. All 240 V plugs have at least 3 and a ground to the plug.
Well no, that's not true. All of the 6-XX series of outlets are 240V with only three wires: Hot1, Hot2, and ground. You are thinking of outlets that need to be dual voltage 120V/240V, like the 14-XX. Those also need the neutral wire, so they can have that 120V between one of the hot lines and the neutral. But a 6-50 or 6-30 is just three wires. In this case, the ground wire is hanging and not hooked up, so it's not currently installed properly, but that can probably be fixed and brought up to code with what is there.
If not, can I run a separate ground wire to the ground rod instead of back to the breaker box?
This can probably be fixed pretty easily. Since the ground wire is in the box already, I would think it does go somewhere useful--maybe comes out right there in your panel where the other two wires go anyway. If you can identify it there, then that would help. So maybe the other end is already tied to a valid ground point, and it may just need to be attached to the outlet. That would be a pretty simple fix then.