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I have found no etymological relationship to "devil" for which the Greek word is "diabolos." The Greek word "daimon" means "demons" and does look similar to Damian, but it is not where the name stemmed from. As this site states under the name Damian, it comes from the Greek name Damianos which was derived from the Greek word "damao" (to tame).
As for the association you speak of, I can only imagine it was derived from the movie/novel "The Omen" in which the son of Satan is named Damien Thorn.