Systematic properties of the AGaH₄ alkali gallium hydrides (A=Li, Na, K, Rb, and Cs) are investigated within density functional theory. Seven ground-state crystal structures are identified, with two energetically indistinguishable structures found for both LiGaH₄, whose structure is as yet undetermined experimentally, and CsGaH₄. Born effective charge tensors, static and high-frequency dielectric tensors, and phonon dispersion relations incorporating longitudinal-optical/transverse-optical mode splittings are computed. Our results indicate that LiGaH₄ and NaGaH₄ have technologically interesting standard enthalpies of formation near −30 kJ/mole H₂. We find, however, that LiGaH₄ is thermodynamically unstable with respect to both LiGa and LiH, providing a possible explanation for its challenging synthesis. The Born stability criteria are evaluated with the computed elasticity tensor components, Cij. All seven structures are found to be both elastically and vibrationally stable.