||A t(2g)(5) system with a honeycomb lattice structure such as Na2IrO3 was firstly proposed as a topological 2g insulator even though Na2IrO3 and its isostructural materials in nature have been turned out to be a Mott insulator with magnetic order. Here we theoretically revisit the topological property based on a minimal tight-binding Hamiltonian for three t(2g) bands incorporating a strong spin orbit coupling and two types of the first nearest-neighbor (NN) hopping channel between transition metal ions, i.e., the hopping (t(1)) mediated by edge-shared ligands and the direct hopping (t(1)') between t(2g) orbitals via dd alpha bonding. We demonstrate that the topological phase transition takes place by varying only these hopping parameters with the relative strength parametrized by theta, i.e., t(1) = t cos theta and t(1)' = t sin theta. We also explore the effect of the second and third NN hopping channels, and the trigonal distortion on the topological phase for the whole range of theta. Furthermore, we examine the electronic and topological phases in the presence of on-site Coulomb repulsion U. Employing the cluster perturbation theory, we show that, with increasing U, a trivial or topological band insulator in the absence of U can be transferred into a Mott insulator with nontrivial or trivial band topology. We also show that the main effect of the Hund's coupling can be understood simply as the renormalization of U. We briefly discuss the relevance of our results to the existing materials.