A first-principles study of the electronic and superconducting properties of the Ni2VAl Heusler compound is presented. The electron-phonon coupling constant of λ(ep)=0.68 is obtained, which leads to a superconducting transition temperature of Tc = ~ 4K (assuming a Coulomb pseudopotential μ(*)=0.13), which is a relatively high transition temperature for Ni based Heusler alloys. The electronic density of states reveals a significant hybridization between Ni-eg and V-t(2g) states around the Fermi level. The Fermi surface, consisting of two electron pockets around the X-points of the Brillouin zone, exhibits nesting and leads to a Kohn anomaly of the phonon dispersion relation for the transverse acoustic mode TA2 along the (1, 1, 0) direction, which is furthermore found to soften with pressure. As a consequence, T(c) and λ(ep) vary non-monotonically under pressure. The calculations are compared to similar calculations performed for the Ni2NbX (X = Al, Ga and Sn) Heusler alloys, which experimentally have been identified as superconductors. The experimental trend in T(c) is well reproduced, and reasonable quantitative agreement is obtained. The calculated T(c) of Ni2VAl is larger than either calculated or observed T(c)s of any of the Nb compounds. The Fermi surfaces of Ni2NbAl and Ni2NbGa consist of only a single electron pocket around the X point, however under compression a second electron pocket similar to that of Ni2VAl emerges Ni2NbAl and the T(c) increases non-monotonically in all the compounds. Fermi surface nesting and associated Kohn anomalies are a common feature of all four compounds, albeit weakest in Ni2VAl.