The self-assembly of DNA-conjugated nanoparticles represents a promising avenue toward the design of engineered hierarchical materials. By using DNA to encode nanoscale interactions, macroscale crystals can be formed with mechanical properties that can, at least in principle, be tuned. Here we present in silico evidence that the mechanical response of these assemblies can indeed be controlled, and that subtle modifications of the linking DNA sequences can change the Young's modulus from 97 kPa to 2.1 MPa. We rely on a detailed molecular model to quantify the energetics of DNA-nanoparticle assembly and demonstrate that the mechanical response is governed by entropic, rather than enthalpic, contributions and that the response of the entire network can be estimated from the elastic properties of an individual nanoparticle. The results here provide a first step toward the mechanical characterization of DNA-nanoparticle assemblies, and suggest the possibility of mechanical metamaterials constructed using DNA.