Data mapping in multiphysics simulation coupling describes the transfer of data between possibly nonconforming meshes. Choosing a numerical approximation method for data mapping is always a trade-off between accuracy and performance. Lower-accuracy methods include, for example, the first-order nearest-neighbour mapping, whereas higher accuracies can often be achieved with a computationally expensive radial basis function interpolation. We extend the multiphysics coupling library preCICE with a greedy approach to radial basis function interpolation. We implement and evaluate the P- and f-greedy methods, which aim to reduce the size of a radial basis function interpolant using a greedy vertex selection approach. The greedy selection is terminated when a user-defined tolerance for a greedy criterion is reached. We compare this method to the nearest neighbour, as well as a global-direct solution and a partition-of-unity approach to radial basis function mapping. We find, that the greedy selection process is computationally expensive for small error tolerances. At the same time, we often see an improvement in mapping time compared to a global-direct solution after the interpolant has been constructed in an offline stage. For high error tolerances, a nearest-neighbour mapping is typically the cheaper option. The partition-of-unity method can achieve comparatively small errors for better runtimes.