DEMMON:

alte red spelling of English and Scottish "Damon."

DAMON:  English and Scottish: from the personal name "Damon" from a classical Greek name, a derivative of "daman" (to kill).  (Compare to Damian).  Respelling of the French Surname "D'amont,"a topographic name, with the preposition "de" denoting someone who lived "a mont" (uphill or on high ground above) a village or settlement.

DAMIAN:

French, Spanish (Damian), Italian (Venice), Czech and Slovak, Greek (Damianos from damazein="to subdue").  St Damian was an early Christian saint martyred in Cilicia in AD 303 under Emperor Domitian, together with his brother Cosmas.  In some accounts the brothers are said to have been doctors, and together they were regarded as the patrons of physicians and apothecaries.  A later St Damian lived in the 7th-8th centuries AD,  and was bishop of Pavia; he may have had some influence on the popularity of the personal name in Italy. Americanized spelling of the Hungarian "Damjan" or the Slovenian "Damjan" or "Damijan."

Jewish (from Poland): either an ornamental adoption of the Polish personal name Damian as a surname, or a habitational name from the village of Damiany in northeastern Poland.

From: Dictionary of American Family Names, Oxford Univ Press, ISBN 0-19-508137-4. Courtesy of Ancestry.com's search tool.