Brinsley is an English surname indicating one's ancestors came from the town of Brinsley in Nottinghamshire, England. According to
A Dictionary of English Place-Names by A. D. Mills, the town's name meant "woodland clearing of a man called Brun".
Perhaps your great-grandparents were fans of the playwright
Richard Brinsley
Sheridan:
http://www.theatrehistory.com/irish/sheridan001.html