Among lightweight materials, magnesium-containing systems are of interest for hydrogen storage. Here, we use an efficient prediction scheme at ab initio level in order to identify new magnesium-based alloys that are stable under elevated pressure. The calculations show that pressure can radically change the minimum energy compositions and various new Mg-rich compounds were found under these conditions. For these Mg-rich compounds, a careful investigation and a search for potential hydrides was carried out and our study on the electronic properties reveals interesting information on the behavior of hydrogen atoms. Our theoretical investigations are in good agreement with very recent experimental results and demonstrate a possibility to identify destabilized metal hydrides. Consequently, the present result encourages the use of similar prediction schemes in order to identify and design destabilized systems for hydrogen storage applications.