The specific role of disorder in the cell-wall material of a two-dimensional cellular solid on its deformation and crushing failure under uni-axial compression was examined numerically and validated with existing experimental data. A random spring network model with nearest and next-nearest neighbour interactions was developed for the representative cellular architecture of staggered-square honeycomb. The springs were taken to be quasi-brittle in nature and to account for the disorder, spring failure properties were taken to be statistically distributed. The model is shown to be capable of reproducing the non-linear behaviour of honeycombs resulting from the damage accumulation and results are shown to be in agreement with experimental measurements in terms of stiffness, strength and energy absorption. Further, the role of disorder in the material properties on the types of failure modes was investigated. Increasing disorder in the spring failure threshold was shown to result in decrease in the peak strength but a more stable crushing response resulting from the increased separation between the two primary cell wall failure modes. © 2021