Lets get back to the original issue. Firstly you have within 18 months to sue. So calm down and wait to see what happens. Don’t get overly excited about a cut.
Secondly, an insurance company will NOT pay out a claim over a slip and fall for a possible scar in the future. You would most likely have to go to court with your child and prove that the property owner was negligent in some way. That would mean that you and your child would have to be there as well as them and their child to explain what happened and how it happened. It is not so simple as you think just filing papers and the insurance company handing you the money. YOU will have to hire an attorney to sue and it could drag out for years. So that will cost you as well. In the mean time as someone else stated, the entire friendship will be ruined and the entire neighborhood will be forced to take sides.
So getting back to what actually happened. Did they do anything wrong? Was anyone there to blame? Could this just as easily have happened at your own home? What if the situation was reversed what would you expect or want them to do?