Nitrogen lines of the N I lambda1134 and lambda1200 multiplets in the damped Lyalpha (DLA) galaxies at z_abs = 2.309, 2.827, and 3.025 toward the QSOs 0100+1300, 1425+6039, and 0347-3819, respectively, have been detected by means of high-resolution spectra (R ~ 2 x 10^4) obtained with 4 m class telescopes at ESO (La Silla, Chile) and the Observatorio del Roque de los Muchachos (La Palma, Spain). The two N I multiplets offer a considerable range in oscillator strengths and the possibility of disentangling Lyalpha interlopers. The derived nitrogen abundances for the three damped systems are [N/H] = -2.68 +/- 0.11, -1.57 +/- 0.09, -2.07 +/- 0.13, respectively. The behavior of nitrogen relative to iron-peak and alpha elements has been investigated by considering all the extant N I determinations for a total of nine DLA galaxies. We have estimated the fraction of iron locked into dust grains to convert the observed [N/Fe] ratios into overall (dust plus gas) relative abundances, [N/Fe]_corr. The ratios [N/alpha] have been mostly determined by using sulphur as a tracer of alpha elements that is unaffected by dust. The [N/Fe] and [N/alpha] ratios show high dispersions, of 1 order of magnitude or more, which have no equivalent in other element-to-element ratios in DLAs. The lowest values of the [N/Fe]_corr and [N/alpha] ratios are at variance with the values measured in Galactic halo stars of similar metallicity, suggesting that part of the DLA galaxies do not follow the chemical evolution of the Milky Way. The DLA nitrogen abundances and their dispersion show some similarities with those observed in dwarf galaxies. Comparison with chemical evolution models shows that the lowest [N/Fe]_corr and [N/alpha] DLA values are close to what would be expected for a pure secondary origin of nitrogen, whereas higher values are mostly consistent with a primary component. The behavior of nitrogen abundance ratios can be ascribed, in general, to the delayed release of nitrogen in the course of evolution. However it is difficult to conciliate this interpretation with the lowest [N/alpha] values measured, since an expected enhancement of alpha elements with respect to the iron-peak elements is not observed simultaneously in these DLA galaxies. In two cases, relatively high [N/alpha] values are observed that require also a more complex chemical evolution to be explained.