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