We calculate the normal-state Nernst signal in the cuprates resulting from a reconstruction of the Fermi surface due to spin-density wave order. An order parameter consistent with the reconstruction of the Fermi surface detected in electron-doped materials is shown to sharply enhance the Nernst signal close to optimal doping. Within a semiclassical treatment, the obtained magnitude and position of the enhanced Nernst signal agrees with Nernst measurements in electron-doped cuprates. Our result is mainly caused by the role of Fermi-surface geometry under influence of a spin-density wave gap. We discuss also possible roles of short-ranged magnetic order in the normal-state Nernst effect and the Fermi-surface reconstruction observed by photoemission spectroscopy.