Models of two-dimensional cellular solids are based on idealized unit cells intended to represent the microstructural features of an average cell from a real material. A significant limitation of the unit cell modelling approach is that it does not account for the natural variations in microstructure which are typical of most cellular materials. Our objective was to model one type of microstructural variability-the non-periodic arrangement of cell walls-and to investigate how this variability affected the relations between microstructure and elastic properties in two-dimensional cellular materials (honeycombs). Specifically, we asked: (1) How much variance in the elastic properties does variability in the arrangement of cells walls introduce? (2) Are the relations between microstructural and elastic properties for non-periodic honeycombs the same, on average, as those for periodic honeycombs? (3) Can anisotropy of elastic properties for non-periodic honeycombs be directly related to microstructural anisotropy, as characterized by stereological parameters? We generated models of non-periodic arrays of Voronoi cells with uniform cell wall thickness, and performed finite element analysis (FEA) to determine effective elastic moduli for low density honeycombs (relative densities < 0.3). Our results indicate that variability in the arrangement of cell walls introduces a modest amount of variance in the elastic constants of isotropic, Voronoi honeycombs (coefficients of variation 4-9%). One consequence of this variance is that differences in the elastic constants in the two directions of nominally isotropic honeycombs are comparable to differences between different honeycombs. Nevertheless, the structure-property relations for both isotropic and anisotropic non-periodic honeycombs are, on average, not different than those for periodic honeycombs. We also measured relative density and microstructural anisotropy using stereology, and demonstrated that the elastic properties of non-periodic honeycombs can be well predicted using this independent estimate of microstructural properties combined with simple relations developed for periodic honeycombs. These findings support the use of the unit cell modelling approach for the analysis of honeycombs of low relative density with random variations in the arrangement of cell walls.