The name is probably a variant of "Linet" used by Sir Thomas Malory in his Le Morte D'Arthur. Linet was a young woman who rode to King Arthur's court to seek help from the Knights of the Round Table for her older sister, Dame Lionesse. King Arthur's youngest nephew, Gareth (then known as Beaumains) undertook the quest. Linet later married Gareth's older brother Gaheris while Gareth married Lionesse.
― Anonymous User 7/8/2009
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I'm not completely sure if the name is of English origin because all other websites list LYNETTE as a Celtic name.