I don't know if this really qualifies, but I would say that Scott Draves can be viewed as an amateur mathematician for inventing/discovering the fractals known as Flame fractals. His work is more towards art, but there is a decent amount of math behind to optimize the aesthetically properties of the fractals. (The "nicest" fractal dimension is 1.52, for example).
Another more recent contribution regards superpermutations, by G. Egan and an anonymous 4chan user. See also this quanta magazine post.