I do not recommend this. It will go no where, be distracting to your true goal and prove to be costly. Merchants have more latitude under the law regarding this sort of "mistake." There are essentially no damages at this point. A claim the store will drop the charge over the threat of a lawsuit is mighty bold considering no one here knows all the facts. That isn't to say innocent or guilty of theft, but whether what the store did was within the bounds of reasonable under tort law. They are permitted to mistakenly accuse people, just not unreasonably so. And there is a fair amount of latitude there.
This is not the Navy where we crush people for speaking up for themselves. In the real world, lawsuits can be levied regardless of how strong they are, and by people in almost any position - including those in jail. If this kid got caught in a chain store, I imagine it was not the company's CEO that was in some back room monitoring the security cameras and calling the cops. This is an isolated incident at the local store's level. Hell, the head manager may not even know it took place. When a lawsuit is levied, you can rest assured that it will reach well beyond the local level, all the way to the headquarters of the business, and I'm certain that they will go back to the local store and tell them to call the local judge and drop this thing. It is just not worth the risk to the company to fight the little guy.
And what's the risk? I'm sure he'd get a deal with a lawyer who would be chompin at the bit to collect potentially huge fees in a settlement.