High-frequency ESR results on the S=1/2 Heisenberg hexagonal antiferromagnet InCu2/3V1/3O3 are reported. This compound appears to be a rare model substance for the honeycomb lattice antiferromagnet with very weak interlayer couplings. The high-temperature magnetic susceptibility can be interpreted by the S=1/2 honeycomb lattice antiferromagnet, and it shows a magnetic-order-like anomaly at TN=38 K. Although, the resonance field of our high-frequency ESR shows the typical behavior of the antiferromagnetic resonance, the linewidth of our high-frequency ESR continues to increase below TN, while it tends to decrease as the temperature in a conventional three-dimensional antiferromagnet decreases. In general, a honeycomb lattice antiferromagnet is expected to show a simple antiferromagnetic order similar to that of a square lattice antiferromagnet theoretically because both antiferromagnets are bipartite lattices. However, we suggest that the observed anomalous spin dynamics below TN is the peculiar feature of the honeycomb lattice antiferromagnet that is not observed in the square lattice antiferromagnet.