We model electronic properties of the second-monolayer Na adatom islands (quantum dots) on the Cu(111) surface covered homogeneously by the first Na monolayer. An axially symmetric three-dimensional jellium model, taking into account the effects due to the first Na monolayer and the Cu substrate, has been developed. The electronic structure is solved within the local-density approximation of the density-functional theory using a real-space multigrid method. The model enables the study of systems consisting of thousands of Na atoms. The results for the local density of states are compared with differential conductance (dI/dV) spectra and constant current topographs from scanning tunneling microscopy.