Scanning tunneling spectroscopy measurements of Pb islands on Si(111) at high energy resolution reveal a novel pseudogap, or a pseudopeak in special cases, around the Fermi level in addition to the usual quantum well states. These gap or peak features persist to temperatures as high as approximately 80 K and are uniquely related to the quantum well nanostructure of the Pb islands. A systematic analysis indicates that electron-phonon scattering is responsible for the observed electronic structure.