Hi Elizabeth , Frances suggestion that Mary could be on target.
When she married John Milburn 18/2/1850 , St George's Douglas , her father (on the IGI) is given as William., one of John & Mary's children was named Robert Casement Milburn , b 1860 , died 1861. Seems they lost a lot of children young. There was also a William Milburn with the second name of ?? , it has been listed with 3 spellings Sowerby, Lowley & Sowrley , I could not find any connection with Thompson/Thomson with these three names in Scotland.
There is also a William & Catherine Thompson in Castletown aged 50 in 1841 and 64 and 63 in 1851 - ?? possible.
Most likely birth in Scotland for Agnes is 10/4/1818 , ch 19/4 Rutherglen , Lanarkshire , to William and Mary Napier , can find a William and John born to this family but no Mary.
On Lawsons there is a burial for Agnes 22/12/1880 aged 62. Also a Will for Robert 1891 , but only information is he left everything to his wife Catherine.
If you search Amanxnotebook , you will find an Obituary for Robert and Agnes son John , who died 23/1/1916 , there is family and historical information.

Cheers
David Dwyer