The structure–property relationship of dielectric elastomers, as well as the methods of improving the control of this relationship, has been widely studied over the last few years, including in some of our previous works. In this paper, we study the control, improvement, and correlation, for a significant range of temperatures, of the mechanical and dielectric properties of polystyrene-b-(ethylene-co-butylene)-b-styrene (SEBS) and maleic-anhydride-grafted SEBS (SEBS-MA) by using graphite (G) as filler in various concentrations. The aim is to analyze the suitability of these composites for converting electrical energy into mechanical energy or vice versa. The dielectric spectroscopy analysis performed in the frequency range of 10 to 1 MHz and at temperatures between 27 and 77 °C emphasized an exponential increase in real permittivity with G concentration, a low level of dielectric losses (≈10−3), as well as the stability of dielectric losses with temperature for high G content. These results correlate well with the increase in mechanical stiffness with an increase in G content for both SEBS/G and SEBS-MA/G composites. The activation energies for the dielectric relaxation processes detected in SEBS/G and SEBS-MA/G composites were also determined and discussed in connection with the mechanical, thermal, and structural properties resulting from thermogravimetric analysis, differential scanning calorimetry, Fourier-transform infrared spectroscopy and X-ray photoelectron spectroscopy analyses.