In the 1880 Census of Raritan, Hunterdon Co., NJ, Jesse lists a Peter Bodine in his household. It says Peter is a brother, but Peter's parents were born in NY and MA whereas Jesse's parents were born in NJ. Peter must be some other relative.

From Ronny Bodine:

In 1900, Jesse Bodine lived in Flemington, Raritan Township, Hunterdon County. He reported that he was a merchant, had been married 38 years and that his wife Mary had borne 2 children, both then living. In 1910, Mary E. Bodine, aged 71 and now a widow, lived with her son William Bodine and his family in Flemington.

Burials in Prospect Hill Cemetery, Flemington, Hunterdon County.
(Find A Grave Memorial # 9788879 & 9788882)
J. P. Bodine 1833-1909 M. E. Whitaker Wife of J. P. Bodine 1838-1913