We investigate realizations of topological insulators with spin-1 bosons loaded in a honeycomb optical lattice and subjected to a SU(3) spin-orbit coupling—a situation which can be realized experimentally using cold atomic gases. In this paper, we focus on the topological properties of the single-particle band structure, namely, Chern numbers (lattice with periodic boundary conditions) and edge states (lattice with strip geometry) and their connection to time-reversal symmetry and the sublattice symmetry. While SU(2) spin-orbit couplings always lead to time-reversal symmetric tight-binding models, and thereby to topologically trivial band structures, suitable SU(3) spin-orbit couplings can break time-reversal symmetry and lead to topologically nontrivial bulk band structures and to edge states in the strip geometry. In addition, we show that one can trigger a series of topological transitions (i.e., integer changes of the Chern numbers) that are specific to the geometry of the honeycomb lattice by varying a single parameter in the Hamiltonian.