1871? RG10, 3777, 136, 20
6 Court, 8 Nelson Street, Liverpool
Boocock, John(?!) Head M 39 Cotton Porter b Manchester
Boocock, Elizabeth Wife F 39 b Manchester
Boocock, John Son M 17 Runner b Manchester
Boocock, Benjamin Son M 13 Errand Boy b Manchester
Boocock, Elizabeth Daughter F 3 b Birkenhead
Looks as if dad changes his name on a regular basis!

Maybe?
Births, December qtr 1867
Birkenhead vol 8a, page 380
Beaucock, Elizabeth (yet another variation!)
There are some trees on Ancestry (Yes! I know how reliable they are!) that have families with:
Thomas Briggs Boocock
Benjamin Briggs Boocock
Elizabeth Briggs Boocock