Turned out to be an issue with an faulty uninstall of Chrome, which left some bits behind in the registry. Bad, bad uninstaller! Go stand in the corner!
Anyway, after numerous searches and a number of unsuccessful fix attempts, Slipstick had the answer. Windows was trying to use the browser that was no longer installed, but was still referenced in the registry.
- Start, click Run..., type Regedit in the Open box, and then click OK.
- Browse to HKEY_CURRENT_USER\Software\Classes\.html
- Right-click the value for the .html key and select Modify...
- Change the value from "ChromeHTML" to "htmlfile" (or from FireFoxHTML to htmlfile)
- Repeat these steps for .htm and .shtml keys if they exist. You may also want to check the .xhtml and .xhtm keys.
Note: As always, the Registry can be a dangerous place, so do a backup/export before making changes.
This didn't even require a reboot and yet another happy customer was made. Judging by the amount of possible issues causing this message, this is only one needle in a field of haystacks.