This article is from the Enya FAQ, by Kevin Martin firstname.lastname@example.org with numerous contributions by others.
From Enya's appearance on the Rosie O'Donnell TV show, 12 Dec 1997:
ROSIE: ...and "Enya" is an Irish name? I never heard [of] it before
ENYA: Where I was born was in the Gaeltact area in County Donegal
and Gaelic is my first language, so "Enya" is Gaelic but there's
a story behind it; there's a goddess in Irish mythology that was
known as "Enya".
ROSIE: And what was the myth about her?
ENYA: Oh, it wasn't so pleasant...
ROSIE: Oh, really?
ENYA: [starts laughing]
ROSIE: Was it that she had a beautiful spiritual voice and then,
ENYA: No. It was actually she was not to marry because it was
destined that her husband would eventually kill her father. So
her father sent her away to this remote island but, alas, she
met somebody and it happened. It wasn't like he intended to kill
the father, but through battle or whatever, the father was killed
by her husband.
ROSIE: Did your father name you?
ENYA: No, it was my grandfather actually. [starts laughing]
[This is a variation on the legend; most books on the subject say it was
Eithne's son, Lugh, who killed Balor, the legendary Eithne's monstrous