This strikes me as a terrible idea, since there are already plenty of laws on the books that make gun owners liable for the misuse of their weapons, either by themselves or other, but they’re asking economists, so Russ Roberts provides the coup de grace:
But the logic is not quite as neat as it might appear. Many people already buy and own guns illegally without license or registration. Adding the cost of insurance would further discourage honest gun ownership. That would make matters worse not better. And is it so obvious that all guns are harmful to others and that gun ownership should be made more expensive to every owner? When an honest law-abiding citizen uses a gun in self-defense, it often protects those nearby who are unarmed. Perhaps gun ownership should be subsidized for honest people. I don’t think this is a good idea, but raising the cost of gun ownership, particularly for good and honest people who are likely to use a gun only in self-defense, is not a free lunch.
People assume that externalities are always negative, which is untrue. I suppose there’s also the problem of deep pockets becoming a target for civil suits.