With recent development of additive manufacturing methods, topology optimization, an increased focus on the generation of designs which maximize material efficiency by lightweighting has gained considerable interest. Lattice structures are one of the popular methods chosen by design engineers for constructing highly complex, functional geometries which are only manufacturable by additive processes. Stochastic lattices have been finding their way into additively manufactured geometries due to their strength at low volume fraction, as well as the ease of implementation with various generative design tools on the market. However, optimization of these stochastic lattices for maximizing part strength and stiffness is a research topic that has been largely overlooked. By tweaking stochastic lattice generation procedures, non-isotropic structures can be generated and these directional strength properties can be exploited. This paper describes a method for homogenizing the effective properties of non-isotropic stochastic lattices generated using stretched Voronoi tessellations, optimization of the stretching aspect ratio and angle within a part design space, and generation of the non-isotropic and smoothly graded Voronoi-based stochastic lattice structures for that design space. The method was applied to a case study of a cantilever beam with nine different Voronoi lattice configurations. Stiffness of parts designed using this procedure was found to be significantly higher than parts designed using an isotropic design.