The stability of magnetic information stored in surface adsorbed single-molecule magnets is of critical interest for applications in nanoscale data storage or quantum computing. The present study combines X-ray magnetic circular dichroism, density functional theory and magnetization dynamics calculations to gain deep insight into the substrate dependent relevant magnetization relaxation mechanisms. X-ray magnetic circular dichroism reveals the opening of a butterfly-shaped magnetic hysteresis of DyPc2 molecules on magnesium oxide and a closed loop on the bare silver substrate, while density-functional theory shows that the molecules are only weakly adsorbed in both cases of magnesium oxide and silver. The enhanced magnetic stability of DyPc2 on the oxide film, in conjunction with previous experiments on the TbPc2 analogue, points to a general validity of the magnesium oxide induced stabilization effect. Magnetization dynamics calculations reveal that the enhanced magnetic stability of DyPc2 and TbPc2 on the oxide film is due to the suppression of two-phonon Raman relaxation processes.