Tailoring materials to achieve a desired behavior in specific applications is of significant scientific and industrial interest as design of materials is a key driver to innovation. Overcoming the rather slow and expertise-bound traditional forward approaches of trial and error, inverse design is attracting substantial attention. Targeting a property, the design model proposes a candidate structure with the desired property. This concept can be particularly well applied to the field of architected materials as their structures can be directly tuned. The bone-like spinodoid materials are a specific class of architected materials. They are of considerable interest thanks to their non-periodicity, smoothness, and low-dimensional statistical description. Previous work successfully employed machine learning (ML) models for inverse design. The amount of data necessary for most ML approaches poses a severe obstacle for broader application, especially in the context of inelasticity. That is why we propose an inverse-design approach based on Bayesian optimization to operate in the small-data regime. Necessitating substantially less data, a small initial data set is iteratively augmented by in silico generated data until a structure with the targeted properties is found. The application to the inverse design of spinodoid structures of desired elastic properties demonstrates the framework's potential for paving the way for advance in inverse design.