From Britain or land of the Britons
Brytnee is a modern creative spelling of Brittany. It originally referred to the Celtic people who settled in the region of Brittany, France, and later Great Britain. This specific spelling variation gained some traction during the late 20th-century trend of replacing 'any' or 'any' endings with 'ee'.
Less common today
Brytnee isn't in the latest US Top 1000. It last peaked at #9,229 in 1991.
US births per 5-year period, 1990βpresent (SSA data).