The "random string" bit is some combination of letters and numbers which is impossible to predict. So.sol: Can be found over at %appdata%\Macromedia\Flash Player\#SharedObjects\\localhost\ Serial.txt: Can be found in the game's steam installation folder, something along the lines of C:\Program Files\Steam\SteamApps\common\the binding of isaac\
I have a Mac and a Windows machine, so I'll go ahead and list the location of said files on both platforms (if anyone has any Linux info, feel free to chip in). The game's save data is contained in two files, both of which need to be moved between computers in order to get everything migrated properly.