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In this paper, natural convection melting in a cubic cavity with internal fins is numerically investigated in detail by employing the lattice Boltzmann method. To have a better understand of the influence of the internal fins on melting process, the configurations of double fins and single tree shaped fin are both considered. In numerical simulations, a parametric study is carried out by varying the fin configuration in the premise of fixed total fins volume, and the influences of the fin configurations on contributing natural convection patterns, melting time as well as the interface area are discussed in detail. Based on the numerical results, it is observed that the existing of fins not only enhances the heat transfer performance, but also maintains similar total amount of stored energy compared with the no-fin case. In addition, the present work also illustrates how changing the fins structures offers a higher rate of heat transfer and a better energy storage capacity rather than increasing the number of fins.