Well, I managed to "find out" about my 3rd great-grandmother. Her actual name is Elisabeth Fineout. It appears, that there is a large group of Fineouts or Finehouts (they tend to add or drop the "h" from generation to generation) in the mid-Hudson region of New York (Dutchess, Ulster, Orange counties) who arrived in the late 1680s. I now need to make my connection to one of the two settlers with that last name.
There are various issues involved with such a search even though I've managed to track down Elisabeth Finehout's parents: John Fineout, b. about 1810 in Scotland - I'm not sure this is actually her father. When dealing with early US Census pages, only the head of household is listed. In the 1850 Census, there is a listing of a daughter Elizabeth born about 1839 which matches my Elisabeth. But I can't find any other info on John Fineout beyond this listing and the 1880 Census listing.
Obviously I have my work cut out for me - I posted a few queries over on the Ancestry bulletin boards and I'll see whats comes of it.