You should probably get a copy of her death certificate which will hopefully indicate how old she was. It sounds like she was born in New York, is that right? If you known when she was born and whether she had brothers and sisters, you'll be able to make a guess at the birthdates of her parents. Then you can try to trace them back to Ireland. Do you know which county? Once you have a date range, you can do a better search.
You should start with the facts that you know. Death certificate, birth certificate, marriage certificate. These all may name parents to give you a start.
It is very possible that the reason for leaving was very personal and so you may not find out. For instance, my great-grandmother left Ireland maybe because she was an orphan. She came to NY when she was 20. Her two aunts lived here, so she may have thought there were more job opportunities here. I don't know, but I can only guess. Previous generations didn't talk much and certainly didn't want to air any dirty laundry.