Thank you for your reply.
Unfortunately, IL does not have a "no pay no play" law, which is a law saying that if someone has no insurance or license they cannot sue for damage in an accident. Thus, even though he did not have insurance or a license, if he was indeed not at fault in the accident, under IL law he is entitled to compensation to pay for his damages. In order to get your mom out of paying for his damages, you would have to sue in court and prove that your mom did not cause the accident and was not at fault.
Even if he had been cited for no insurance and no license, if your mom was at fault in causing the accident, she would still be liable for his damages. In fact, even if he had insurance, if your mom was at fault she would have had to pay for his damages.
The only way your theory would legally be correct would be in a state with a "no pay-no play" law which is where a person without insurance would not be able to sue anyone for damages resulting from an accident. Unfortunately, that is a minority of states and IL is not one of those.
In addition, whether or not someone is cited for no insurance or no license is not considered a "fault citation" and as such would not have bearing on civil liability for the accident I am afraid.
So if you want your mom to get out from liability for the accident, you have to go to court and prove she was not at fault for the accident.