As is typical with trying to locate a place named after a pit/mine colliery, there are sometimes complications.
Barrington Colliery consisted of at least three pits/shafts. The Hannah Pit ( sunk 1854), the Henry Pit and the Molly Pit.
The Hannah Pit, a "green field" site, and the community that grew up around it, lay just south of Stakeford, just south of the River Wansbeck.
The Henry Pit and the Molly Pit lay just south west of the community of West Sleekburn.
The Durham Mining Museum web site
www.dmm.org.uk , in the Barrington Colliery section, has helpful map showing both Barringtons, sandwiching Bomarsund and West Sleekburn, on a Ordnance Survey map ( 1860 edition)
In today's terms this would be in line with the modern-day road A1147.
Trivia Bit. The pub, the Lord Barrington exists today in Stakeford ( or is that corner in Bomarsund ? )
Michael Dixon