I'm not a fan of either:
Eileen is a family name, but I wouldn't use it. I probably prefer
Aileen, except that my friend's mother was born into a family that gave all the children A- names; she was named
Aileen but they pronounced it
Eileen. (Friend is
Antoinette; her brother is
Alan.) I regard both
Aileen and
Eileen as dated, and a two-year-old one would amaze me.