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The advent of additive manufacturing has enabled the precise manufacturing of lattice materials at different length scales. While honeycombs are usually constructed from a tessellated array of identical unit cells, most structures in nature tend towards stochastic or disordered tessellation. In this work, we explore the effect of disorder properties of honeycombs, in particular, stiffness, strength, fracture toughness, and crack propagation paths using finite element models. The honeycombs are created using Voronoi tessellations with different levels of disorder measured by a regularity parameter δ , where δ = 1 corresponds to a regular hexagonal honeycomb and decreasing values of δ correspond to increasing levels of disorder. We consider three constituent materials: very brittle, quasi-brittle, and very ductile (0.0007–0.25 in failure strain). The results show that pseudo-order realizations with 0.5 ≤ δ ≤ 0.8 show the best strength-toughness trade-off. With increased plasticity, the improvement in fracture toughness compared to regular honeycombs can be as high as 50% at the expense of a 20% decrease in strength. This increase in fracture toughness is achieved both at crack initiation with crack blunting intrinsically and extrinsically during crack growth by a combination of crack deflection and crack bridging. Also, a single mathematical variable δ usually referred to as a "regularity parameter" to control cell stochasticity cannot account for the variability in tensile strength and fracture toughness parameters for brittle or quasi-brittle constituent materials and significantly more for a ductile constituent material. Thus, a local parameter for disorder is required to explain the large statistical variance seen in the mechanical properties of disordered honeycomb structures across the same irregularity value. [ABSTRACT FROM AUTHOR]