Jack,
A search on RootsChat for Lamberton brings up several posts - a couple include further information about available records.
The IGI lists 9 children between 1858 and 1873 to Robert GIBSON and Ann RENWICK, all born in Berwickshire Scotland (the first in Duns, the second in Hutton and the rest in Coldstream). Do you have the
birthplaces of Robert and of Ann from censuses? Presumably they were in Scotland for the 1861 and 1871 - these censuses are online at (the pay-per-view site) ScotlandsPeople at:
http://www.scotlandspeople.gov.ukIf you find their birthplaces, you might be able to find them in the 1841 and 1851 censuses (on SP if they were in Scotland) and this could help you to identify their parents.
I assume that neither of them died in Scotland? Scottish death certificates (also available on SP) are wonderfully informative and include (if known by the informant) the the name and occupation of the deceased's father, and the name and maiden surname of the deceased's mother.
JAP