You can mark it either red or black, but you already have one black wire, so it would be best to mark it red, just to differentiate it.

Since you went with 10/2 for a 240v outlet that means your cable will have two hots and a single ground. But the cable you bought will have a black, white and bare/green wire. Make sure that you wrap both ends of the white conductor with red electrical tape to indicate that it is also a hot wire. White normally indicates a neutral, but since you won’t have a neutral wire in this cable, it’s important to mark it. Otherwise someone else may work on this outlet in the future, see a white wire, assume it’s a neutral, wire something up with it as a neutral, and give themselves a pretty bad shock.

If you’re in North America, which I assume you are based on the 120v wires, then your assumption about 240v power is a bit off. Both wires can be a hot (120v) wire, 180° out of phase from each other, so they add to 240v. In this case, a neutral wire isn’t necessary to carry the current back, the other hot wire does that. A neutral wire may be used, but then there would be 4 wires.

The ground wire and the neutral wire actually connect to each other in the panel, but it’s not safe to use a ground wire in place of a neutral, so definitely don’t wire it to the neutral on the outlet.

If you are unsure of what’s going on with this set of wires, you should really call an electrician to help. Wiring a standard 120v outlet is something a homeowner can do, but identifying an old 240v cable on a dubious circuit is definitely something a qualified electrician should do.