<quote from Wikipedia> Lafayette was born at the Château de Chavaniac, near Le Puy-en-Velay, Haute-Loire, in the Auvergne region of France also known as the "Appalachia of France" for the remoteness and mountains. He belonged to the cadet branch of the La Fayette family, which had received its title ("La Fayette") from an estate in Aix that belonged to the Motier family in the 13th century. The original Gilbert Lafayette (from whom Lafayette draws his motto, "CUR NON", (Latin for "WHY NOT") fought at the Battle of Baugé (also called Battle of Beauge) and later for Joan of Arc. His father was killed at the Battle of Minden in 1759 by a British cannon ball, and his mother and grandfather died in 1770. He was educated by his Aunt and 2 priests (the 2nd was the Abbe Fayon, Cure de Saint-Roch de Chavaniac) He married at 16 to Marie-Adrienne-Françoise de Noailles, daughter of Jean-Paul-François, 5th duc de Noailles. She simply called herself "Adrienne" or "Noailles Lafayette" and became widely known for her simplicity, extraordinary charity, and bravery, both well before and during the French Terror. <unquote>
It is quite an old name. Sorry I could not find the etymology.
Wise Owl: I searched around the obvious analogy "la faille" => "la faillette" but could not find anything so definitive. Thanks.