In this Study we use density functional theory (DFT) to investigate the properties and H adsorption characteristics Of Structural isomers of the magnesium-doped aluminum Cluster, Al12Mg. Our results show that the exohedral structure (exo-MgAl12) is signisignificantly lower in energy (1.59 eV) than the endohedral structure (endo-Al12Mg); however, the exohedral structure shows significant structural distortion. Our calculations demonstrate that H binds favorably to both exohedral and endohedral structures. Generally, binding energies for H to both clusters range from 2.3 to 2.5 eV with atop positions slightly favored, except for addition directly to the exohedral Mg atom, where the binding energy drops to 1.92 eV. We include a DFT molecular dynamics study of the endo-Al12Mg and endo-Al12MgH clusters which revealed the isomerization to the respective exostructures at finite temperatures (100-600 K). Interestingly, hydrogen adsorption appears to enhance the isomerization.