Hi Elaine! I am related to Anna-her father and my great-grandfather were brothers.
I believe that it is possible that the "H" in William H. does stand for Harrison, but I am not certain. He had a nephew who was named William Harrison. My grandmother told me once that she believes she remembers hearing somewhere that her uncle, William Harrison, was named after her great-uncle William H.
As far as I know, no one in our family that I have talked to knows for sure if the "H" truly stands for Harrison...it is just a suspicion.