We report neutron diffraction studies on a series of La2NiO4+delta single crystals with 0.05 less than or similar to delta less than or similar to 0.11. At 300 K, all of the crystals have an average tetragonal structure (space group I4/mmm). On cooling below 290 K, one or more orthorhombic phases appear, characterized by incommensurate superlattice peaks at (0, k, l +/- DELTA) with k odd, l even, and 1/4 < DELTA less-than-or-equal-to 1/2. The positions, widths, and intensities of the superlattice peaks are quite sensitive to the cooling rate. We show that the incommensurate peaks are evidence of intercalated layers of oxygen spaced periodically along the c axis, with a one-dimensional ordering similar to the staging of intercalates in graphite. The structure of the La2NiO4 lattice between the interstitial layers is of the Bmab-type; the superlattice peaks result from the ordered autiphase domain boundaries induced by the interstitial oxygens, which sit at (1/4, 1/4, 1/4)-type position.. Observed orderings involve interstitial layers separated by two to four Ni-O layers. Peak shapes and positions are modelled quantitatively using the formulas of Hendricks and Teller for one-dimensional disorder in a layer lattice. Besides the one-dimensional ordering of the intercalant layers, temperature-dependent phase separations are observed. Because of the slow ordering kinetics, phase separation can be suppressed by rapid cooling.