We present a first-principles study of the equilibrium geometries, electronic structure, and related properties (binding energies, ionization potentials, electron affinities, and magnetic moments) of free-standing Ni-n (n = 1-10) clusters doped with one impurity of N. Calculations have been performed in the framework of the density functional theory, as implemented in the SIESTA code within the generalized gradient approximation to exchange and correlation. We show that, in contrast to the molecular adsorption of N-2, the adsorption of a single N atom can dramatically change the structure of the host Ni-n cluster, examples of which are Ni5N, Ni7N, and Ni10N, and that noticeable structure relaxations take place otherwise. Doping with a nitrogen impurity increases the binding energy as well as the ionization potential (except for Ni6N), which proves that N-doping works in favor of stabilizing the Ni clusters. We also find that the magnetic moments decrease in most cases upon N-doping despite the fact that the average Ni-Ni distance increases. The HUMO-LUMO gap for one spin channel strongly changes as a function of size upon N-doping, in contrast with the HUMO-LUMO gap for the other spin channel. This might have important implication in electronic transport properties through these molecular contacts anchored to source and drain electrodes.