From Britain or Brittany
A modern phonetic variant of Brittany, referring to the Celtic region in northwest France or the island of Great Britain. It gained some usage in the late 20th century following the popularity of the original spelling.
Less common today
Britne isn't in the latest US Top 1000. It last peaked at #3,014 in 1990.
US births per 5-year period, 1990βpresent (SSA data).