Stuart,
You are asking for suggestions about a cheap rifle, in single shot, that has history and character. I can't think of a single one.
The only cheap single shot rifle is the Handi-Rifle, and it does not have history or western character.
I believe that the cheapest single shot rifles with history or character are going to be modern reproductions of the old plains rifles, and they are going to be in the $700 to $1,500 range.
These would include modern reproductions of the Sharps, the Browning Hi-Wall, and the Remington Rolling Block. I am sure that other folks on this board can provide additional examples.
Now, if you are willing to go black powder, then you can probably buy a modern reproduction of the .50 caliber Hawkin, plus a good set of peep sights, for under $500. But, you are limiting yourself to 150 yards or so, and hunting only in good weather.
The Ruger No. 1, in any of a dozen classic calibers, would also fill the need. But, I think you are again looking at $700 plus (used), and that doesn't include the peepsights or scope. The Ruger is not a classic single shot rifle, but it is a modern interpretation of the classic single shot, and can be an excellent rifle.
Just my thoughts.
Mannyrock