Copper with high thermal and electrical conductivity had been widely utilized as frictional materials. However, their wear resistance was too weak to meet the actual requirements. In this study, graphene nanosheet and hexagonal boron nitride (h-BN) with strong mechanical properties were coated onto the surface of copper to improve the wear resistance. Their nano-tribological properties were studied by molecular dynamics (MD) simulation to understand the wear mechanisms well from the nanoscale. The simulated results indicated that the average friction coefficient of copper coated by h-BN and graphene decreased by almost 50% compared with pure copper. The interactions between Cu atoms and carbon atoms in diamond counterpart and graphene were interpreted by analyzing the variations of the radius distribution function (RDF). The h-BN and graphene lubrication mechanisms were discussed by monitoring van der Waals (vdW) force, slippage, the energy of the system, and temperature variations during the process of friction. Furthermore, Von Mises stress distributions on the copper substrates and dislocation evolution were also investigated to virtually observe the change of atoms.