To sum, having her pay out of pocket might be your best option. Let me explain.
Since it sounds like she is responsible and doesn't have insurance, she would ultimately have to pay out of pocket for the damages one way or the other. With that being said, if your son has uninsured motorist property damage coverage this would also be an option to have the bumper repaired. Uninsured motorist property damage would pay in the event that someone hits you without insurance and a hit and run. There may be a deductible for your uninsured motorist coverage. Also, keep in mind filing a claim (even when you are not at fault) can affect your insurance rates. The rates can be affected because either the company had to pay a claim and/or there is a higher risk of an accident in the area you are driving. If you do not have uninsured motorist, but have collision coverage, you can file a claim under collision. If you do this, the accident will show as at fault but the damages will be repaired minus your deductible. Many companies do require you to report an accident even if a claim is not made. Make sure to check with your specific company for their rules.
If the driver of the other car is admitting fault and you do not want to file a claim or have the coverage to file a claim with, having her pay out of pocket would likely be the best option. You can also consult an attorney if they do not pay for the damages. Good luck.