Electron-phonon interaction in the Si(111)-supported rectangular phases of In is investigated within the density-functional theory and linear-response. For both single-layer and double-layer structures, it is found that the phonon-induced scattering of electrons is almost exclusively determined by vibrations of In atoms. It is shown that the strength of electron-phonon coupling at the Fermi level λ(EF) increases almost twofold upon adding the second In layer. One of the reasons is that additional low-frequency modes appear in the phonon spectrum, which favors a strong enhancement of λ(EF). The agreement of the calculated parameter λ(EF) = 0.99 for a double-layer structure as well as the superconducting transition temperature Tc = 3.5 K with experimental estimates indicates that the discovered superconducting phase is probably a double-layer rectangular -In structure on Si(111) with a coverage of 2.4 ML. This conclusion is also supported by good agreement between the calculated electron band structure and ARPES measurements.