SBC,
In the 1831 census there was a Samuel Black in Craigmore, in Aghadowey (close to Lisboy). His family was Presbyterian and contained 5 males and 2 females. Possibly John Black’s father:
http://www.census.nationalarchives.ie/pages/1831/Londonderry/Coleraine/Aghadowey/Craigmore/16/
Griffiths Valuation for Movenis in 1859 lists a Jane Bradley on plot 5C(b). That was a labourer/weaver’s cottage. I’d suspect that it could be Jane Bradley's widowed mother. This might possibly be her death:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1879/06532/4882251.pdf
Ardreagh is a mile or so north of Movenis. Jane is in the Valuation revision records for Movenis till 1869 when she left. She died in Ardreagh but does not appear in the Valuation records for that townland. However her 1879 death was reported by Thomas Millican and he does appear in the Ardreagh records. He had plot 9b (another weaver’s cottage) so I’d guess she was lodging with him. I think Thomas’s wife Mary Anne was Jane’s sister, or sister - in – law.
Thomas in 1901:
http://www.census.nationalarchives.ie/pages/1901/Londonderry/Aghadowey/Ardreagh/1515819/
Thomas & Mary Anne’s daughter’s baptism which gives the mother’s maiden name as Bradley:
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1869/03388/2242009.pdf
They married in 1861:
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1861/09607/5483850.pdf
The families seem to have used Aghadowey Presbyterian church. It’s baptisms only start in 1855 and marriages in 1845 (though the church was founded in the 1600s). Copy of those records is held in PRONI.
There are at least 3 Black gravestones in Aghadowey Presbyterian church graveyard. Whether they relate to your family is hard to say. Weavers, bleachers, labourers often couldn’t afford a gravestone and so would have been buried in unmarked graves. No Bradley gravestones there as far as I know.
The families being Presbyterian and living in the Aghadowey area point strongly to them being descendants of Scots who settled there in the 1600s. Many came from Ayrshire. Speaking of his youth in Aghadowey in the 1830s, page 25 of Autobiography of Thomas Witherow 1824 – 1890, pub 1990 by the Ballynacree Historical Society:
“Aghadowey had originally been settled by a Scotch migration and I found that my new neighbours spoke as pure Scotch as a man might hear in any part of Ayrshire.”