Hi Kelly
Here is the NZ Immigration Passenger List for the "Bebington" arrived at Wellington October 1872.
https://www.familysearch.org/ark:/61903/3:1:S3HT-DYJ7-98W?i=16&cc=1609792Your HANCOX family are listed as "HANCOCK" ... see the entry at Image # 17.
[Underneath that is the HANCOX family of Henry(24), wife Sarah and children. Was Henry related to James ? ]
Often you can find a list of births (and deaths) aboard a vessel, listed in the last pages of the passenger list. Unfortunately can't find that on the "Bebington" list.
Another source may be the publication in a newspaper of the passenger list / ships arrival at a port, where there's a summary of the voyage which includes say, the number of births (and deaths) on
board - though names are not always noted.
Someone has already mentioned here the death in 1877 of an Albert Edward HANCOX ? That death was registered at Tapanui,
Marlborough (Province) *Otago. Is that where your family settled ??
[I haven't yet checked PapersPast for the death ... nor have I seen the passenger list on rootsweb.]
~ Lu
Edited : To correct Province, which is incorrectly given as Tapanui,
Marlborough (source: NZ Deaths @ancestry.com)
It should read as Tapanui, Otago.