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.

Roger Lustig
Princeton, NJ USA
Research co-ordinator, GerSIG

Evertjan. wrote:
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".