My guess is that the maker is not "Benjamin Franklin," but rather the Benjamin Air Rifle Company which (at least as of 1941) used to be based in St. Louis, Missouri. The reason I said, "...at least as of 1941..." is that I have a copy of Outdoorsman magazine from January of that year, and it includes a box ad by Benjamin listing that address, and featuring information about its air rifles and pistols which could shoot BBs or pellets (with rifled barrels in two available calibers: .177 or .22).
Benjamin is now owned by Crosman. For information on your particular heirloom pistol you might check Crosman's Website (www.crosman.com). Click on "About Crosman," then "Customer Service," then "Product Support," and finally "Benjamin/Sheridan Date of Manufacture."
There really are "Benjamin Franklin" airguns (as opposed to Benjamin or Benjamin Sheridan airguns). I don't know much about them, but they're out there.
I did a bit more research and found that the "Benjamin Franklin" brand of air guns (pistols and rifles) were, indeed, manufactured by the Benjamin Air Rifle Company of St. Louis, Missouri. Crosman acquired the Benjamin marque in 1991.
Yes, thay made a bucn of 'em marked "Benjamin Franklin" over the years. That pistol is probably a model 132(.22) or 137(.177). There are people out there who still fix them. Here's a coupla places to ask:
the crosnam forum: http://www.network54.com/Forum/275684