I'm replacing an old sub panel in a small mother-in-law apartment. The wiring from the main panel does not have a ground wire. The old panel was grounded to the water heater which sits on concrete blocks. What is the best way to ground the new main lug box?

The feeder to this panel should be a four wire, with an insulated neutral, and a bare or insulated equipment ground, keeping the neutral and EQG separated.

The neutral bar is easily insualted and separated, you may need to add an equipment ground bar both directly to panel using machine thread screws.

Yes it should be but the bare wire is missing. Just the black, red, and white. Changing out the wire is not an option. Is there another option.

What type of cable or raceway is there?

Changing a panel will require the feeder to be brought up to current code.