We theoretically demonstrate that the desired -orbital honeycomb electron lattice can be readily realized by arranging CO molecules into a hexagonal lattice on a Cu(111) surface with scanning tunneling microscopy (STM). The electronic structure of the Cu surface states in the presence of CO molecules is calculated with various methods, i.e., density functional theory (DFT) simulations, the muffin-tin potential model, and the tight-binding model. Our calculations indicate that, by measuring the local density of states (LDOS) pattern using STM, the -orbital surface bands can be immediately identified in experiment. We also give an analytic interpretation of the -orbital LDOS pattern with the method. Meanwhile, different from the case of graphene, the -orbital honeycomb lattice has two kinds of edge states, which can also be directly observed in STM experiment. Our work points out a feasible way to construct a -orbital honeycomb electron lattice in a real system, which may have exotic properties, such as Wigner crystal, ferromagnetism, -wave superconductivity, and the quantum anomalous Hall effect. Furthermore, we also propose a simple way to calculate and identify the modified Cu surface bands in the Cu/CO systems with the DFT simulations. Considering the recent works about a -orbital square lattice in similar systems [M. R. Slot , Nat. Phys. 13, 672 (2017)1745-247310.1038/nphys4105; L. Ma , Phys. Rev. B 99, 205403 (2019)2469-995010.1103/PhysRevB.99.205403], our work once again illustrates that the artificial electron lattice on a metal surface is an ideal platform to study the orbital physics in a controllable way.