Dictionary.com says a Gyre is a "ring or circle". The book so far has dealt with Bruce accepting a new vigilante in Gotham and Silver St. Cloud in his personal life, so it seems to refer to Batman embracing new people in his own personal circle.
As an aside, it also says in terms of oceanography, a gyre is "a ringlike system of ocean currents rotating clockwise in the Northern Hemisphere and counterclockwise in the Southern Hemisphere". Since most of this issue takes place on an island, it could kinda hint at something else.