We explore by computational modeling the effects of boron-nitrogen (BN) composition on the thermal and mechanical properties of amorphous silicon-boron-nitride (Si-B-N), a synthetic ceramic material with superior thermal protection, mechanical attributes, and oxidation resistance at high temperatures. Network-derived Si-B-N models optimized with ab initio molecular dynamics serve as input structures for classical molecular dynamics simulations. Atomistic Green-Kubo simulations on relaxed supercells and structural relaxations on strained cells are used to screen the thermal and mechanical properties of a collection of network structures with low enthalpies of formation. We find that when the material is composed of well-mixed parts rather homogeneously spread within the material, the thermal conductivity and elastic constants are isotropic and exhibit a weak dependence on composition and network structure. In contrast, when separation into BN-rich layers occurs, the material exhibits anisotropic behavior, with an increase in thermal conductivity along the layer direction and decrease in elastic constant in the cross-layer direction. The insights provided into the composition-structure-property relationships can be useful for the rational design of amorphous Si-B-N materials targeting high-performance coating applications.