"Cockcain" proves to be a very unusual variant of the name more often seen as Cockayne and similar. The whole of Ancestry yields just four matches:
- Samuel Cockcain, who paid 2s 8d in tax to Fannett township, Pennsylvania, in 1786.
- Mary Cockcain, buried at Sheering, Essex, on 2 September 1790
- Thomas Cockcain, buried at the same place on 13 March 1785 (quite possibly Mary's husband)
- Reginald Cockcain, freemason, who at the age of 35 was inducted into the Greta Lodge (Keswick, Cumberland) of the United Grand Lodge of England on 14 October 1919.
None of these seem relevant here.
A similar global search on FindMyPast (similar to Ancestry but more UK/Ireland oriented) yielded just one hit, a 1910 passenger list from New South Wales containing a "Mr and Mrs Cockcain" (no further details in the summary, and the original is in part of the site I don't have a subscription to).
I'll try and dig a bit more using variants of the name.