With finite bandgaps, g-BNC, a boron nitride monolayer (g-BN) phase within a graphene layer, is a promising semiconductor for next generation electronics. We report its mechanics dependence of the g-BN concentration, including the high order elastic constants and mechanical failure, through a first-principles study based on density functional theory. The in-plane stiffness as well as third order elastic constants of graphene can be linearly tuned with g-BN concentration. The longitudinal mode elastic constants are sensitive to the BN modification, in contrast to the shear mode elastic constants. This study may provide guidance in optimizing the mechanics of graphene-based nanodevices.