The pronunciation has of two variants. Aileen is the Scottish spelling of the name Eileen, and was originally pronounced the same (I-LEAN). People of Scottish decent still pronounce it this way. The name is also pretty common in Latino cultures, where it is also pronounced I-LEAN. Whites of non-Scottish European descent are the ones who say A-LEAN.
This is my name and my family comes from the far western side of Ireland. The name has been in our family for generations and we pronounce it neither A-Leen or I-leen, but like "AL-een or AL-leen" almost.
A- soft A, like the beginning of alligator
Actually basically leave off the 'igator' and replace it with the 'een' sound.
Usually I just tell people it's like Ali/Allie/Ally with an N at the end.
I know it's not the typical way of pronouncing it, but we've been doing so for generations.