MindS1 is correct that in terms of electrical properties you don't need to be terribly picky. As you noted you will want multicore wire to allow for flexibility and prevent breakage.
I'd also add that the standard 9V battery boxes are current limited, so in practice you can get away with 22 AWG or even 24 AWG wire. I'd encourage you to try out the wire in advance to make sure it meets your power requirements. A pair of buggy motors is very different than a few lights or a train motor.
It is possible, though more difficult, to replace the entire wire rather than splicing out and replacing the bad bits. This is covered in detail on Instructables. Here's the gist:
- Carefully remove the back plate from the connector using two flat blade screwdrivers to release the clips:
- If you are using a compatible wire size, you can simply slide the wire into the metal clips exactly as the original wire was attached. Otherwise, you can solder it in place:
- Pop the back plate back on being careful to route the wires and tabs appropriately.
If you are working with the longer wires, and the break is near the end, you can simply remove the wire, cut off the bad end, and reconnect the remainder of the original wire. Your wire will be a few cm shorter, but that isn't a big deal for many applications.