The death certificate is online for Hugh born 10/10/1859 and died 5/20/1913 showing parents John and Mary Fulton Sheridan...https://familysearch.org/pal:/MM9.3.1/TH-1942-21165-45629-82...
There is a Hugh Sheridan, age 10, in household of Mary Hamilton in 1870 census in Athens Co. Mary appears to be head of household at age 50, and also listed are another Mary Hamilton, age 28, Hugh Sheridan 10, William Lewis 2 and Walla Hamilton 2 months.
There is a marriage license for a Hugh Sheridan, Jr., and a Mary Cooper in Athens Co, OH, in 1887. Since he calls himself "Jr", it could possibly be assumed that his father was also Hugh, and this may not be the same person as for above death certificate. You can see the license here:https://familysearch.org/pal:/MM9.3.1/TH-1-18059-94216-18?cc...
The 1900 census in Trimble, Athens Co, OH, shows Hugh 39, Mary F 36, Walter 16, William 10, Elizabeth 7 and Martha 5.
The 1910 census in Waterloo, Athens Co, OH, shows Hugh 50, Mary 49, William 20, Lizzie 17, Martha 15, Glayd 6, Agnes 4, and Oscar 1.
There are marriage licenses for Oscar, Walter, Lizzie and Martha on family search that show father as Hugh Sheridan and mother as Mary Cooper, as well as some death certificates.
The confusing part of this information is the "Jr" on the marriage license, and the father stated as John on the death certificate. That would need some investigating to be sure these are the same people.
Hope this helps somewhat..