hi there, the only thing i can see is on the IGI there is a christening 6 june 1847 st peter liverpool for a james bedson whose parents were down as james bedson and mother sarah although it does not give her maiden name.
also on the 1881 census parkgate, great neston, cheshire there is a family as shown below
james bedson head   m     aged 71   born leighton cheshire       fisherman
charlot bedson wife    m     aged 68  born leighton
james bedson son              aged 28  born leighton                      gardener
george bedson  son           aged 26  born leighton                       fisherman
thomas bedson son           aged 24  born leighton                       fisherman
mary e edge  grandaughter     age 9  born barnston
don't know if any of this helps
regards
belinda