We investigate the thermoelectric properties of Fe/MgO/Fe(001) magnetic tunnel junctions (MTJs) by means of the linear-response theory combined with a first-principles-based Landauer-Büttiker approach. We find that the Seebeck coefficient of Fe/MgO/Fe(001) MTJs strongly depends on the barrier thickness and the tetragonal distortion. A compressive tetragonal distortion of the in-plane lattice parameter in the MTJs provides interface resonant states just above the Fermi energy. This causes resonant tunneling in the MTJs and significantly enhances the Seebeck coefficient when the thickness of the MgO barrier is around 1 nm (four or five atomic layers of MgO). Moreover, an extensive tetragonal distortion of the in-plane lattice parameter pushes the interface states away from the Fermi energy, leading to a reduction of the Seebeck coefficient. Furthermore, we find that the interface resonant tunneling enhances the power factor of the MTJs for the compressive distortion. These results indicate that control of the barrier thickness and the tetragonal distortion will be effective for maximizing the thermoelectric properties of MTJs.