Designing magnetic shape memory materials with practicable engineering applications requires a thorough understanding of their electronic, magnetic, and mechanical properties. Experimental and computational studies on such materials provide differing perspectives on the same problems, with theoretical approaches offering fundamental insight into complex experimental phenomena. Many recent computational approaches have focused on first-principles calculations, all of which have been successful in reproducing ground-state structures and properties such as lattice parameters, magnetic moments, electronic density of states, and phonon dispersion curves. With all of these successes, however, such methods fail to include the effects of finite temperatures, effects which are critical in understanding how these properties couple to the experimentally-observed martensitic transformation. To this end, we apply the quasi-harmonic theory of lattice dynamics to predict the finite-temperature mechanical properties of Ni-Mn-In magnetic shape memory alloy. We employ first-principles calculations in which we include vibrational contributions to the free energy. By constructing a free energy surface in volume/temperature space, we are able to evaluate key thermodynamic properties such as entropy, enthalpy, and specific heat. We further report the elastic constants for the austenite and martensite phases and evaluate their role as a driving force for martensitic transformation.