That's great that A. Brown's grandfather, John Hamilton Brown actually referred to the Liverpool Wylies as cousins.
This implies that their mother, Mary Wylie was possibly a sister of the Browns in L'derry. And also then that Amy J Brown, (John Hamilton's half sister) married her cousin (Uncle Sandy) in Liverpool.
The Browns in L'derry as far as I know were:
John (b1847 in Johnstone, Renfrewshire, d1897 Derry)
Alexander (b1849 in Johnstone, d1929 Derry)
James (b1851 in Johnstone, d1933 Derry)
Robert Hamilton (b c1858 in Derry, d?,mysterious disappearance), A Brown's great grandfather
Janette (b1860, d?)
Agnes Stewart (b1862 Derry, d1946 Portstewart)
Margaret (b1867 Derry, d1953 Belfast) my great granny
Jean (b1869 Derry)
I think there was also a son, David Hepburn Brown (Bertie), born I think in Derry, who went to South Africa.
Their father was Alexander Brown, born c1818 in Ayrshire, married in 1845 in Renfrewshire to Agnes Stewart, b 1824 in Renfrewshire.
Alexander Brown was previously married and widowed between 1841 and 1845, possibly in Newton upon Ayr, Ayrshire.
From the first marriage he had atleast one daughter, Eliza, born c 1844 in Newton.
((The 1851 census in Johnstone, Renfewshire gives him and his second wife Agnes with Eliza aged 7, and then John aged 3 and Alexr aged 1. James must have been born shortly afterwards and then the family , for some reason, went to Derry.))
What I don't know is how the sister Mary Brown/Wylie, (Alex Hampton's great grandmother) fitted into this. From the Liverpool census' it can be seen that she was born in c1843 in Ayrshire, so either she is "Eliza", or she is another sister just not in the Scottish 1851 census.
Alex, I went to the Central Library in Liverpool and was able to find that the Wylies are buried in Kirkdale Cemetery in Section 7, Grave 1668. Well, I found Amy Johnstone Brown/Wylie, who died in 1912 aged 22, and John Wylie, your great-grandfather, who died in June 1917, aged 79. I really want to know who else is in the grave. Particularly to find when Mary Wylie died, and then if there is any more info on her. This will show if indeed she is a daughter of Alexander Brown. And then who her mother is.
((To get this info, I have to contact the Liverpool cemetries' office to obtain the Order Numbers of each burial for the grave, and then return to the library to look up each Order Number.))
Sorry for the huge response!