I don't have an actual answer, but here is a (somewhat remote) possibility:
On the 1880 census, I find a Minnie Lee Grim, parents Wellington (close, sort of, to William) and mother Mary C. (possibly Mary Catherine?). Data follows...
1880 United States Federal Census
Name: Minnie Lee Grim Home in 1880: Ashby, Shenandoah, Virginia Age: 7 Estimated Birth Year: abt 1873 Birthplace: Virginia Relation to Head of Household: Daughter Father's Name: Wellington Father's birthplace: Virginia Mother's Name: Mary C. Mother's birthplace: Virginia Race: White Gender: Female
Household Members: Name Age Wellington Grim 45 Mary C. Grim 45 Annie E. Grim 16 Minnie Lee Grim 7 Charles E. Grim 6 Ada F. Grim 2 Henry Peirson 77 Casper Baker 22