Surface properties are examined in a chiral d-wave superconductor with hexagonal symmetry, whose one-body Hamiltonian possesses intrinsic spin-orbit coupling identical to the one characterizing the topological nature of the Kane-Mele honeycomb insulator. In the normal state, spin-orbit coupling gives rise to spontaneous surface spin currents, whereas in the superconducting state, besides the spin currents, there exist also charge surface currents, due to chiral pairing symmetry. Interestingly, the combination of these two currents results in a surface spin polarization, whose spatial dependence is markedly different on the zigzag and armchair surfaces. We discuss various potential candidate materials, such as SrPtAs, which may exhibit these surface properties.