Since this site focuses on the Civil War, if you know her dad's name, that would
actually help you. More than likely he fought in the Civil War and he honestly
could have been either Union or Confederate.
Since the lady was born in 1861, that is after the census of 1860, and it would be 1870
before there would be a chance for her to appear on a census. But you truly need
a surname and a location to make progress. Things shifted so much in the whole
nation between 1860 and 1870, it is all shifting sand. However, I would say do
not be discouraged, as I have learned things as much as 25 to 30 years
after I started the search.
Go to your local library and see if they have a genealogy section or someone
who works that as their specialty. Perhaps they can help point you in the right
direction. That would be a great place to start.
Good luck.