In this work, we address a problem of recovering a boundary condition on an elastic cavity from a single boundary measurement on an external part of the boundary. The boundary condition is given by a Robin condition and we aim to identify its Robin coefficient (matrix). We discuss the uniqueness question for this inverse problem and present several numerical simulations, based on two different reconstruction approaches: An approach by solving the Cauchy problem and an iterative Newton type approach (that requires the computation of several direct problems). To solve the mentioned (direct and inverse) problems, we propose the Method of Fundamental Solutions (MFS) whose properties will be discussed.