I presume only card up to 2GB work with minimig, maybe some cards up to 4GB. The 2GB limit is typical due to FAT16 limitations, but the real border between SD and SDHC is at 4GB. I doubt there are many cards available that are more than 2GB, and that are not SDHC.
Adding support for SDHC sounds like an obvious upgrade possibility for the minimig :-)