My girlfriend bought a house with a detached workshop/shed. The shed had existing electricity but doesn't connect to the house in any other way (no phone, no pipes, etc.)

The existing wiring in the shed had several things wrong (including a single cartridge fuse shutoff with about 5 circuits all wired into it!)

So I ripped all that out, installed a 6 breaker subpanel with separate ground bar, etc. I'm pretty skilled with wiring. My Uncle was a licensed electrician and when I was in high-school I wired our house when it was built. (My Uncle would come on Saturdays - inspect my week's work and give me lessons for the next week's work).

Here comes my question...

The existing underground wire from the house is only a 12-2 NM. That is wrong for two reasons. NM vs UF (see other recent discussion). But the 12-2 doesn't have a ground. So now I have a subpanel without an equipment ground.

We are planning on replacing the underground wire at some point (with something bigger and with a ground), but in the meantime, I'd like to ground the subpanel. There are no water pipes, etc. in the shed. Can I install a ground rod at the shed?

As it is, all you have a 20A 120V feeder. If you were still working under pre-2008 codes, you could bond the neutral and ground at the subpanel, install a ground rod (or two) and be OK, but you only have 20A to work with and only every other breaker in your panel will work (unless you jumper the two legs).

I've already jumpered the two legs in the subpanel. All we have are single pole breakers (three 20A; three 15A) and we're careful about only using one tool at a time.

We'll try to win the lottery but until then I'll pick up a ground rod.

If your existing buried NM is old enough that it doesn't have a ground, it may be about on its last legs. I've pulled up buried NM with ground where the jacket was literally peeling off. That NM had failed shorted in a lightning storm - a surge finished it off.

It will require a grounding electrode system even if fed with a four wire service.
The grounding electrode system is not for personnel safety but for lightening and pole transformer high voltage leakage.
The earth rarely has a low enough impedance to trip even a 15 amp breaker if a 120 V hot line is connected to the earth.

The now banned 'Worm Getter' relied on this.
Two electrodes were driven into the earth a few feet apart, then connected to 120 V.
The electricity irritated the earth worms and they fled to the surface to be collected.

The problem is that people trying to gather the worms often failed to disconnect the electrodes and got electrocuted trying to pick them up.

Sub-panels in separate structures look a lot more like a main panel, since that is what they actually are FOR THAT STRUCTURE.