Hannah's name in America was Hannah Lee. In Norway she was Hanna Hansdatter daughter of Hans Nilsen from Ringsaker in Hedmark and Ingeborg Pedersdatter of Faaberg. Hanna was born in Hage, Faaberg parish in Oppland. How the family acquired the name Lee I have not figured out. Usually, with Norwegians, this could be expected to be a place name to which Hans was connected but I cannot see such a place in his history in Norway.
I have gathered up transcriptions of Hans Nilsen family in the 1865 and 1875 census as well as images of the parish records pertaining to the family (Hans & Ingeborg's marriage and the christening of Hanna and her siblings).