We have employed parameter-free density functional theory calculations to study the thermodynamic stability and structural parameters as well as elastic and electronic properties of Ni4N in eight selected crystallographic phases. In agreement with the experimental findings, the cubic structure with Pearson symbol cP5, space group Pm3m (221) is found to be the most stable and it is also the only thermodynamically stable structure at T = 0 K with respect to decomposition to the elemental Ni crystal and N2 gas phase. We determine structural parameters, bulk moduli, and their pressure derivatives for all eight allotropes. The thermodynamic stability and bulk modulus is shown to be anticorrelated. Comparing ferromagnetic and nonmagnetic states, we find common features between the magnetism of elemental Ni and studied ferromagnetic Ni4N structures. For the ground-state Ni4N structure and other two Ni4N cubic allotropes, we predict a complete set of single-crystalline elastic constants (in the equilibrium and under hydrostatic pressure), the Young and area moduli, as well as homogenized polycrystalline elastic moduli obtained by different homogenization methods.We demonstrate that the elastic anisotropy of the ground-state Ni4N is qualitatively opposite to that in the elemental Ni, i.e., these materials have hard and soft crystallographic directions interchanged. Moreover, one of the studied metastable cubic phases is found auxetic, i.e., exhibiting negative Poisson ratio.