Given a closed manifold, do the areas of minimal surfaces determine the metric uniquely? Does the Dirichlet-to-Neumann map for the Laplacian determine the metric? These questions are part of a growing field of geometrically and analytically natural inverse problems. Inspired by work in the compact setting, we will discuss minimal surfaces in asymptotically hyperbolic spaces and a corresponding "renormalized" area that is conformally invariant. We show that knowledge of the renormalized area on a relatively small subset of minimal surfaces determines the asymptotic expansion of the metric, including the conformal infinity. As a further application, we show that renormalized area can determine the conformal structure of the boundary of a hyperbolic 3-manifold.