You really do have a tough one on your hands.I've searched mostly the 1860 and 1870 Census records with both first and last names, and with Hinckley spelled various ways (Hinkley, Hinckly, etc.).
This is what I found in only the 1870 Census:
A "Anna Hinckley" age 13, is boarding with McLay, Harrison and family. in Ridgeville Twp, Lorain County, Ohio.This Census was taken on August 22, 1870.
Then, also in Ridgeville Twp., staying with the residence of Peck, Harisson and Lydia, is Eliz (Eliza?) Hinckley, age 42, Anna Hinckley age 14, and Marietta Hinckley, age 12.There is also a Terry Maer, age 22, with this same household.This Census was taken on August 15, 1870.
The dates of the Census are included because that may be helpful in determining which Anna is yours, if either are.
Then in 1880, Avon Twp., Lorain County, OH:
ROOT Henry E. age 23 Anna M. age 23 Carlton H. age 2 Florence M. age 10/12