When in doubt, check Lars Menk's A Dictionary of German-Jewish Surnames.
He reports that the name means "person >from Flehingen."
The earliest examples he has used both FLEHINGER and FLEGENHEIMER.
FLEHENHEIMER was also known.
Princeton, NJ USA
Research co-ordinator, GerSIG
Flegenheimer seems to me an inhabitant of the theoretical
stetl "Flegenheim", though ShtetlSeeker cannot find it
which has, in German, nothing to do with "flat".