The extension of data-driven computational mechanics to finite strain elasticity is validated with different types of manufactured material datasets, using a Lagrangian finite element formulation of the problem and a novel resolution scheme for the minimization problem subject to nonlinear constraints. It was found that both a rather sparse database, containing only the three standards tests used to fit constitutive models for elastomers, and an optimal database computed from numerical heterogeneous simulations on a 2D structure, are able to predict the global response of another 2D structure. The local strain and stress states are better captured by the optimal dataset, at a very reasonable cost for the combinatorial problem (only 2,200 strain-stress couples in the database), suggesting that adapting the material database to the problem is the best way to perform data-driven simulations.