An eikonal model is presented and evaluated for the elastic scattering of Li-11 from target nuclei. The Li-11 is treated as a Li-9+n+n three-body system interacting with the target through two-body Li-9- and neutron-target interactions. A careful quantitative investigation of the sensitivity of the calculated elastic-scattering cross section to the required theoretical inputs, the neutron and Li-9-target potentials and the assumed structure of the Li-11, is presented. Since the structure of the Li-11 halo is not yet known, we investigate the extent to which elastic-scattering data are able to discriminate between different theoretical Li-11 wave-function models. We find that there is some sensitivity in the calculated elastic cross-section angular distributions to the degree of neutron correlations in the Li-11 wave function and that these are more significant at lower incident energies and on lighter targets. We conclude that the current uncertainty in the Li-9-target interaction is the major obstacle to elucidating the structure of Li-11 from present elastic-scattering data.