Enhancing the efficiency of thermal storage systems is important and necessary. Numerical research on investigating the effects of new fins, magnetic fields, and nanoparticles on the melting of phase change materials (PCM) in a finned cavity is presented in this paper. Effects of fin and its shape, porous fin, adding nanoparticles, the non-uniform magnetic field, and the oscillating magnetic field on melting of PCM are evaluated to determine the state that has the most thermal performance. The enthalpy-porosity method is employed for simulating the melting process. The results specify that the melting time in finned mode with circular, triangular, rectangular, sin-downward, and sin-upward shapes decreased by 44%, 46%, 51%, 53.9%, and 54%, respectively, compared to the no-fin mode. The liquid fraction is increased to 98% by adding nanoparticles. By employing the porous fins, the liquid fraction is decreased by increasing the porosity coefficient and Darcy number. For the sin-upward fin, the liquid fraction is increased by 54%, 48.29%, 22.99%, and 9.50% using a hybrid nanoparticle/non-uniform magnetic field, nanoparticles, porous fin, and non-uniform magnetic field, respectively, compared to the base case. The highest melting performance belongs to using the sin-upward fins with nanoparticles and a non-uniform magnetic field.