What to do if my son was in a car accident and the insurer wants him to sign paperwork stating that he will pay $10,000 in damages?

Presumably, if the insurer wants your son to commit to paying $10,000, there are something like $10,000 in damages at stake, and at least some chance your son was at fault in causing them. Therefore, your son should consult with an attorney in detail about this situation before doing anything at all.

As a matter of law, your son would be liable for the actual cost of any damages or personal injuries (and potentially "pain and suffering" for serious peronal injuries) he caused if he was at fault. A driver is typically at fault if he was driving negligently, or unreasonably carelessly: driving too fast for conditions or violating the speed limit; DUI/DWI; texting or talking on the cell phone while driving; going through a red light or stop sign; etc.

The fact that the insurer wants your son to pay does not mean he was in fact doing any of these things or otherwise driving carelessly. But again, they presumably would not be asking unless it was at least possible he was. Your son needs experienced counsel to review the situation with him to determine whether it appears it he was in fact at fault and what his liability, or financial exposure, might be. No one should ever sign anything which might obligate them for $10,000 without speaking with an attorney.