The Western Normal Newport was named after the Western Association Normal School Masters of Dancing, a professional organization for dancing masters in western North America. “Normal school” sounds odd to modern ears, but historically, it didn’t mean what we’d imagine today. A normal school was a teacher-training school, where the “norms” of teaching and subject material were taught; more information on normal schools in general may be found here. The Western Association Normal School was founded in 1894 under Canadian dancing master John Freeman Davis, its first president.
My only source for the Western Normal Newport is George Washington Lopp’s La Danse (Paris, 1903). Lopp was from the northwestern USA and may have been a member of the Normal School or at least aware of it. The variation’s name, as given in French by Lopp, was Le Western Normal Newport, attributed to l’Association de l’École normale des Maîtres de danse de l’Ouest.
Given that the school was founded in 1894, the Western Normal Newport was obviously created too late to appear in M. B. Gilbert’s Round Dancing, published in Portland, Maine, in 1890, from which Lopp cribbed so much of his own book.