W Briarcliff Lane runs about 400 feet west from 39th Avenue W before it makes a 90° turn to the north and becomes Briarcliff Lane W. That segment runs about 200 feet north to W Dravus Street. A private road, it is part of the Briarcliff development on what was once Briarcliff School (1949–1984, demolished 2003). You can read more about the history of Briarcliff at HistoryLink.com and on the website of the Magnolia Historical Society.
As for the school’s name? In an article published August 21, 1948, The Seattle Times noted that “on recommendation of district residents, the board named the new school being built in the Magnolia area the Briarcliff School.”