I disagree that the song ruined the name. Songs don't really do that. The homophone with "I lean" is more annoying than the song association.
I'm not really a fan of names that stress the second syllable the way Eileen does. And I have issues with its spelling. The EEN is dated IMO. 70 years ago an EEN name probably looked/sounded charmingly exotic but not too "ethnic" ... to me it looks super-midcentury-Americanized, like 'look, I'm not an immigrant.' Anyway now you can spell it Eibhlin (with marks =P) even if you aren't Irish. Or heck, they're naming girls Ireland now. lol
But I think -een probably will not look frumpy to the generation born today so if I really liked Eileen that much myself I'd totally use it for a baby.