“Marriage Records of Highland County, Ohio (1805-1880)” by David and Jane McBride, copyright 1962, gives 6 joseph West marriages but only one to an Elizabeth:
Joseph West and Elizabeth Turner on 18 Aug 1844 by Sampson Tener. He was Justice of the Peace in Brush Creek Twp. and also a minster of the Univeralist Church.
1870 census shows them in Brush creek with sons Owen and Marion and a daughter Hephzibah. “Wills, Administrations, Guardianships and Adoptions of Highland County, Ohio (1805-1880)” by David and Jane McBride, copyright 1957. Contains abstracts from ProbateCourt records. On page 284, 6 Aug 1874, Elizabeth J. West was appointed Guardian of her children Marion (19) and Hepsa Jane (14), heirs of Joseph B. West, deceased. He must have died between the 1870 and Aug. 1874.
I can't tell you who his parents were but “Highland Pioneer Sketches and Family Genealogies” by Elsie Johnson Ayres, copyright 1971, says that John and Joseph West were early pioneers of Bush Creek. They were born in the mid 1700's and had a brother named Owen who settled in nearby Clinton County. Just a possibility.