Plumbyone...hello
Some thoughts....
I think this might be Elizabeth Hill in 1871 with family before her marriage in 1877
https://familysearch.org/pal:/MM9.1.1/V513-VF3 You can see a reference in the household to a 3yr old William Barnfather which is promising given you have the surname coming up later.
The best way to be sure about anything is just to try and confirm a couple of events via certs (I know costs build up) but you have Samuel Gordon and Elizabeth Ballantyne Hill's marriage in Carlisle, 3rd Qrt Vol 10b, pg 756 (likely you have this already). I would guess that Ballantyne (don't worry about the spelling of this) is probably just Elizabeth's middle name.
This is the 1881 census entry you referred to earlier
https://familysearch.org/pal:/MM9.1.1/XQST-B74Cannot easily see Lillian's birth on
http://www.freebmd.org.uk/cgi/search.pl in 1881 but wondered whether Mary rather than J for a middle name was actually 'I'? There is a birth reg in Carlisle for a Mary Imrie Gibson - 1879 3rd Qrt Vol 10b, pg 500.
There is a couple showing on Family Search, a John Gibson and Mary Imrie, having children from 1855 (and the start of official Scottish Registration) up to 1859: Isabella 1855, David 1857 and David 1859 (likely first David died)...and then nothing...maybe a move to England?
I am late and have to rush, will look again later.
Monica