I'd turn it in to the cops. They have the ability to determine ownership; I don't.
The sentiment is proper to a degree, the ability of the local cops to find the owner is completely absent in most cases.
They would have to contact the ATF who could run a serial number check to see where the maker sent it, and at the first stop then its all on them. Distributor located in another state 15 years ago? Ok, contact them, on the taxpayers dime while attempting to keep up with real crimes and stuff happening on a daily basis. Days later the distributor gets back to them via the extremely slow email process which is hiding in their inbox filled to capacity, and you discover it was shipped to another state not bordering yours. Contact that retailer to discover it was shipped to a LGS in a third state and sold to someone there.
There is no national database of who owns what. That is the point. At best it finally goes thru an FFL where he searches a literal bound book of transactions - if he hasn't gone out of business and forwarded those records to the ATF, and now sitting in a warehouse next to a large crate with Nazi markings singed off and other curiousities from Area 51 hiding it from view.
About the best they can do is search a database of known stolen gun serial numbers and if no hits come up, then it's a dead end. Dude lost his gun, old boy is losing it, best for him and the family if he's incapable of keeping track of his stuff.
The best recourse, if you think the gun fairy didn't leave you a present, is to leave your number with no identifying comments about you or the gun in question so those who lost it can at least give you some credible proof it's theirs. EXPECT MORE THAN ONE CALLER. It's going to be like craigslist buying your used car and offering a low ball price. "You found my gun, great, where can I meet you I can be there in 15 minutes I have a reward see you there!" and hang up.
Keep in mind no good deed goes unpunished and watch your 6.
Or they could just do nothing and then if desireable they bid on it at a quiet auction selling off property no longer needed by the jurisdiction.