Most unfortunately Rich Harzog, the owner of Exonumia.com (the site linked by Quant-Geek) and the Exonumia Press, died in November last year. I don't expect the site will be up and running much longer.
With a little luck, you may find something with the search "a f barney" site:archive.org in Google. There are thousands and thousands such punches, as you can see in the four lists. And yes, most are professionally made punches. They were a convenient way to advertise oneself as a blacksmith, merchant, hotel owner, etc. etc. at the local level and as long as the coin circulated locally.
Note that new punches are found regularly. I have quite a few myself that remain to be published.
G.G. Brunk in Merchant and Privately Countermarked Coins lists on page 350, entry B-312, an 1867 U.S. two cent piece with the countermark of D.P. Barney. This may be totally unrelated to your coin. No photo is provided for that coin.
There is also entry B-313 with the countermark of E.H. Barney on an 1854 U.S. half dollar.
The coin is low grade, and it looks like the date is worn out. If you can read the date, compare with the values here.
With the countermark, since A.F. Barney has not been identified yet, I would expect the selling price be about $7 to $15 or, if you're very lucky, $20. If you could identify him with a reasonable degree of probability, perhaps $30 to $35? The price will vary greatly according to how and when and to whom you want to sell it.
In short, if you can't read the date, it is basically worthless to a collector of decimal coins. With the counterstamp, you might grab the attention of a collector of such items, even more so if you can identify the issuer.
Information I found A.F. Barney {Albert Frances Barney} from Laconia, New Hampshire was an inventor.
In July 21, 1896 he got a patent for a "blind operator and lock" # 564,180
Nice research, Ken. You're doing the right thing, but:
Two-cent coins were struck only until 1873, so one would expect someone active in the 1870s. It may be the case with Albert, but it would need to be demonstrated.
A question that arises is "Why would an inventor countermark coins?" I suppose he wasn't an inventor full time, i.e. that he also had a trade or profession. He may have been a locksmith, blacksmith, hardware retailer, or something like that (or whatever is the kind of trade where someone invents a "blind operator and lock") which involved him or his employees handling cash.
If so, his name may well show up in other documents, such as trade journals which were normally published monthly or even weekly. I have tried these searches: "albert f barney" site:archive.org (no results); "a f barney" site:archive.org (some results, but most/all seem too late); I suggest also "barney albert f" ... "barney a f", etc. The same could be done in Google Books. I select those sites otherwise you get hundreds of thousands of results in a regular search.
It has happened to me as well to get people listed as patent owners, but so far, none has ever had the right profile for a countermark under further examination. Good luck anyway!
The coin is dated 1864. It was worn before the countermark. That would mean it was stamped later. Even 1900 would not be out of line. I would think A.F. Barney maybe had a hardware store. His patent was for a lock on windows.
Good point, the coin circulated a long time, though I doubt many 2 cents were still circulating in the late 1890s. But there is a surprisingly high number of 2 cents which have been countermarked in this way, so it may well be that, as they were withdrawn from circulation, they were reused as cheap and convenient metallic business cards if you punched them with your name. The late 19th and early 20th centuries was a period when coin-shaped business cards (often limited to the name of the office or business and some basic services offered or items sold) were quite popular.
