  We theoretically demonstrate that the desired $p_{x,y}$-orbital honeycomb
electron lattice can be readily realized by arranging CO molecules into a
hexagonal lattice on 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, \textit{i.e.}~DFT simulations,
muffin-tin potential model and tight-binding model. Our calculations indicate
that, by measuring the LDOS pattern using STM, the $p$-orbital surface bands
can be immediately identified in experiment. We also give an analytic
interpretation of the $p$-orbital LDOS pattern with $k \cdot p$ method.
Meanwhile, different from the case of graphene, the $p$-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
$p_{x,y}$-orbital honeycomb electron lattice in a real system, which may have
exotic properties, such as Wigner crystal, ferromagnetism, $f$-wave
superconductivity, quantum anomalous Hall (QAH) 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
$p$-orbital square lattice in similar systems [M. R. Slot, \textit{et al.} Nat.
Phys. \textbf{13}, 672 (2017); Liang Ma, \textit{et al.} Phys. Rev. B
\textbf{99}, 205403 (2019)], our work once again illustrates that the
artificial electron lattice on metal surface is an ideal platform to study the
orbital physics in a controllable way.
