I love threads like this.
Daf, KwikFit have no contractual relationship with your son, or the numpty he allowed to drive his car without insurance. Kwikfit have a contractual relationship with the other driver.
Your son may have a claim in law against the other driver. He needs a lawyer who understands motoring law to pursue that claim. My guess is that without a conviction against the other driver or prima facie evidence (rear end shunt for instance) most lawyers are going to be pessimistic about going to court with this one. It's only balance of probabilities, but your son starts from behind the eight ball in terms of persuading the court that he's telling the truth. Will a lawyer do this on a conditional fee basis? I wouldn;t bet on it, but happy hunting.
Your son may also have a claim in law against his friend, the non insured numpty, but probably only if he didn't know that the numpty had no insurance.
Good luck.