A detailed analysis of the quantum inelastic dynamics involving the lower rotational levels of the MgH^- (X ¹ Σ⁺) and MgH⁺ (X ¹ Σ⁺) molecular ions in collision with He atoms in a cold trap is presented. The description and comparison of the potential energy surfaces between the molecular partners and the He (¹S) gaseous atoms is followed by the quantum, multi-channel dynamics carried out. By considering a range of temperatures which are expected to be present in a cold ion trap experiment, the multi-channel kinetics of population changes is simulated for the molecular ions during the collisional cooling process. The rotational populations finally achieved at specific temperatures are linked to the feasibility of the present anion MgH^- to be involved in state-selective photo-detachment experiments.