We demonstrate the compositional effect on the magnetodynamic and auto-oscillation properties of Ni 100 - xFe(x)/Pt (x=10-40) nanoconstriction-based spin Hall nano-oscillators. Using spin-torque ferromagnetic resonance performed on microstrips, we measure a significant reduction in both damping and spin Hall efficiency with the increasing Fe content, which lowers the spin pumping contribution. The strong compositional effect on spin Hall efficiency is primarily attributed to the increased saturation magnetization in Fe-rich devices. As a direct consequence, higher current densities are required to drive spin-wave auto-oscillations at higher microwave frequencies in Fe-rich nanoconstriction devices. Our results establish the critical role of the compositional effect in engineering the magnetodynamic and auto-oscillation properties of spin Hall devices for microwave and magnonic applications.