If a person were to miss a few days of running because of an injury or illness, is there any benefit of running small doubles to make up some lost mileage? I don't want an answer that says just recover first from whatever took you out, but I'd want to know if there are actually any benefits to doubling in this situation assuming the runner is healthy again.
It would probably benefit you mentally. That in turn sets you up to get back at it which obviously would benefit you all around.
Are you saying doubles instead of the same mileage in singles? If so, then does it come down to a question of doubles being better to do while recovering from your sickness/injury in order to get the same volume in?
No. The standard advice in training books is that you should not try to "make up" mileage due to injury, illness, travel, etc. You risk over-training, fatigue, injury, etc. by making up mileage. Pick up where you left off and "program in" rest days for the future.