In depth knowledge of the thermal and rheological properties of the crust and mantle of Mars is fundamental for constraining the strength of its lithosphere, and thereby for the understanding its geodynamics, tectonics, and thermal evolution. In this context, an interesting debate on Martian crustal composition is ongoing. The presence of highly differentiated rocks at several locations, and their implications for the thermal state, structure, and evolution of the planet are not well understood. Here we systematically explore the effects of crust material properties (specifically thermal conductivity and density) on the thermal and mechanical structure of the Martian lithosphere. For this purpose, we analyze different key indicators of the thermal state, the strength and mechanical behavior of the lithosphere under a wide range of conditions. We do so by considering suitable parameters for both a nominally basaltic Martian crust and an end-member basaltic crust that would include a significant low density and high thermal conductivity component. We find that crust material properties have a strong control over the thermal state of the entire lithosphere, and thereby over the strength of the lithospheric mantle. Although a lower crustal density reduces the brittle strength, the colder geotherm due to a higher thermal conductivity leads to a stronger crust and lithospheric mantle, and therefore to a thicker lithosphere. It also leads to a stronger lithosphere as a whole in terms of total strength and effective elastic thickness, as a consequence of higher crust and mantle contributions. On the other hand, we also investigate the influence of the rheology of Mars' upper mantle on the total strength of its lithosphere. Water content has a large effect on the rheology of the upper mantle. A wet rheology implies a substantial reduction of the mantle contribution to the total strength and effective elastic thickness of the lithosphere, resulting in a significantly weaker lithosphere as a whole. Our results will serve both to improve our understanding of geophysical observations from the InSight and ExoMars missions, and to further constrain theoretical modeling efforts.