MathWord's article on Lucas circles traces the name to a little-known 1973 publication. These interesting circles have found their way into several 21st century publications, including the online Encyclopedia of Triangle Centers. Is there a reference to Édouard Lucas's discussion of these circles?
This is what Paul Yiu writes about the origin of the name:
Éduoard Lucas (1842-1891) wrote about this triad of circles, considered by an anonymous author, as the three circles mutually tangent to each other and each tangent to the circumcircle at a vertex of ABC. The [construction in terms of] inscribed squares was found by Victor Thébault (1883-1960).
Thébault's paper is summarized here. So it seems the name "Lucas circles" goes back to this 1937 source.
I presume Lucas's source is his Récréations mathématiques (1882-1894), but I have not yet located it there.