From Britain or land of the Britons
Bryttney is a modern creative spelling of Brittany. The name originally referred to the region of Bretagne in northwest France, which was settled by Britons fleeing the Anglo-Saxon invasions of England. It gained immense popularity in the United States during the 1980s and 1990s, leading to many phonetic and decorative spelling variations.
Less common today
Bryttney isn't in the latest US Top 1000. It last peaked at #10,007 in 1990.
US births per 5-year period, 1990βpresent (SSA data).