The Nederlandse Voornamenbank (First names bank of the Netherlands) says
Edme or Edmée is a French name, derived from
Edmond. There is a Dutch singer who is called Edmee (prn. ed-MAY).
http://www.meertens.knaw.nl/voornamen/VNB/index.php?act=zoeken&id=4578
When I looked for genealogy websites I found a lot of French families, but also this:
http://familytreemaker.genealogy.com/users/a/l/e/Boyd-Alexander-Occoquan/WEBSITE-0001/UHP-1780.htmlI don't know if he was Scottish, but he died before 1650.