• An efficient design framework for foam-like structures is newly presented. • The stochastic nature of Voronoi structures is considered in the proposed framework. • The optimized Voronoi structures show 95.25% improvement in efficiency compared to the cases where the proposed framework is not implemented. • The proposed methodology is a powerful tool for efficient design of Voronoi structures for a specific target objective. Recently, many studies have increasingly focused on developing bio-inspired structures, leveraging their lightweight and high-energy absorption properties, which are crucial across many engineering fields. Structural optimization aiming for bio-inspired structures having superior energy absorption capability, however, has been considered a challenging problem. One of these challenges is that nonlinear material behaviors induced by external forces, such as buckling and self-contact of constituting ligaments, intervene in the energy absorption process. Such nonlinearities not only make the relationship between design changes and energy absorption nonlinear, but also exacerbate the difficulties of design, given the complexity of the ligament configurations. To address this, a novel design optimization method for bio-inspired cellular structures with high energy absorption is proposed. First, Voronoi tessellation is used to capture configurations of bio-inspired material, parameterized by geometric variables. Then, Bayesian optimization with Kriging efficiently updates the design, exploring the complex design space through high-fidelity nonlinear finite element analysis. The proposed design method is efficient in structural optimization as it combines a strategy to reduce the number of samples required for surrogate modeling of structural response and optimal search, but it also generates multiple design outcomes with similar advantages due to the intrinsic variance of the Voronoi structures.