We present a detailed study of the dielectric and charge transport properties of the antiferromagnetic cubic spinel HgCr2S4. Similar to the findings in ferromagnetic CdCr2S4, the dielectric constant of HgCr2S4 becomes strongly enhanced in the region below 60-80 K, which can be ascribed to polar relaxational dynamics triggered by the onset of ferromagnetic correlations. In addition, the observation of polarization hysteresis curves indicates the development of ferroelectric order below about 70 K. Moreover, our investigations in external magnetic fields up to 5 T reveal the simultaneous occurrence of magnetocapacitance and magnetoresistance of truly colossal magnitudes in this material.