Thank you for your response and yes, I have visited that web page a few times.
I've tried the variation spellings of her name but she's not there. If she came to New York via Ellis Island wouldn't she have travelled with her family (her husband, and son). I found a Eugene DORSET on ElllisIsland.org and my GGgrandmother's husband was Eugene DORSETT and he travelled with a small boy. Why wouldn't they all be together? I'm sure you don't know the real reason, I'm just thinking out loud. Not sure if that's the right Eugene.