My grandmother was adopted in 1918 and she, and her 2 siblings, kept the names given to them by their birth mothers. I sent off for her birth certificate which gave me her birth mother's name and address. It turned out that her adoptive parents were related by marriage to her birth family.
If however your great grandmother was given a new name, then it will be much more difficult to locate her birth parents' details.
Maybe there are clues in the census, or perhaps she was baptised by her birth mother before being handed over.
Good luck with your search.