This is an example of a general topic called self-rescue. There are textbooks on the subject: Reference request: books on self-rescue while climbing .
The simplest situation would be where the arm was injured before you started the rappel (as opposed to getting hurt while doing the rappel), and the person with the injury doesn't feel confident about being able to rap safely with the uninjured arm. In that situation, I think the easiest solution would be to have your climbing partner lower you using his belay device for friction. This would be just like lowering someone off of a sport climb, but the belayer would be above rather than below the person being lowered.
As an alternative, the injured person might feel safe enough to rap off with one arm if there was a solid safety backup to the rappel setup, e.g., a Prusik backup: Backup prusik for Abseiling: Where does it go? . One concern would be that when things go wrong in a rap, the solution is often to ascend the rope with a friction hitch or a mechanical ascender -- but a person with a broken arm might not be able to do this.