In this study, body-centered-cubic Ti24V18Nb6Cr12 high entropy alloy (HEA) is selected as the matrix to design Mg–Ti–V–Nb–Cr lightweight HEAs. Based on first-principles calculations, the hydrogen storage properties of a series of Mg–Ti–V–Nb–Cr HEAs are systematically investigated. The designed Mg-based HEAs are found to be stable after hydrogenation and exhibit high gravimetric hydrogen storage capacities (HSCs). Particularly, the gravimetric HSC of Mg6Ti24V18Cr12 HEA is as high as 4.091 wt%. Besides, the dehydrogenation temperature of the Mg–Ti–V–Nb–Cr HEAs decreases obviously with increasing Mg content, which results from the weakened bonding strength of <M-H> pairs. This study demonstrates that the Mg6Ti24V18Cr12 HEA is a potential candidate for hydrogen storage.