My Uncle Merl researced the name Emeigh and determined that it started in Italy and was derived from Amache. The family moved to Germany and then to France. Eighmey, Aimee, and Amei are just a few of the different spellings. My Uncle indicated that the original family was comprised of clothing mamkers. As was the custom in ancient times, the workers chose the guild's name as their surname. The coat of arms included a spider and web to signify clothing manufacturers. My Uncle completed his research approximately 30 years ago when he was in his 40's.
Another point of interest is that there were Emeigh's on both sides during the American Revolution. The loosers changed the spelling of their names and some moved to Canada.