That's a real scream, Pete! (Sorry, I've been rooted at my desk all day and am dripping sarcasm...)
A banshee actually is Celtic in origin. Bean is Gaelic for woman and the Sidhe (pronounced 'shee') is the name given to the fairy folk. The banshee is a woman fairy who attends the most famous or most important families (usually lords and kings) and upon their death, she sings a death wail. 'Sing' is used freely here. I believe it was the basis for the Klingon death cry: to alert the otherworld that an important person was coming.
(Guess I'm also dripping trivia.) Time to unroot and make dinner...