Mary Emily Richards was one of the "orphan train" adoptees. The 1910 census lits two Mary E Egan's, both 13, both adopted, both born in New York. One of course is Eleanor M (or Mary Eleanor), and the other is Mary Emily. Her birth name was Richards.
Though not genetically an Egan or a Rooney, even though her "great grandfather" was Michael Rooney on the Padua side, she married William Edward Rooney whose great grandfather was John Rooney the Avon side. Thus by prior relationship they were 3rd cousins, but by consanguinity they were unrelated.