Is this just a 1-off, or are you in volume manufacture?

For a 1-off, you could easily put a unique ID into the code.

For manufacture, you probably want just a single code build - the unit's Serial Number would be independent of the software version. Especially if you ever need to do software updates...

If you don't like Robert's suggestion with the Dallas silicon serial number, you could have an EEPROM on the board, and program a serial number into that as part of your Final Test in manufacturing...