Rare-earth indium oxides RInO3 (R = Gd, Tb, Dy) consist of spin-frustrated hexagonal spin lattices made up of rare-earth ions R3+, where R3+ = Gd3+ (f(7), L = 0), Tb3+ (f(8), L = 3), and Dy3+ (f(9), L = 5). We carried out DFT calculations for RInO3, including on-site repulsion U with/without spin-orbit coupling (SOC), to explore if their low-temperature magnetic properties are related to the two nonequivalent nearest-neighbor (NN) spin exchanges of their hexagonal spin lattices. Our DFT + U + SOC calculations predict that the orbital moments of the Tb3+ and Dy3+ ions are smaller than their free-ion values by similar to 2 mu(B) while the Tb3+ spins have an inplane magnetic anisotropy, in agreement with the experiments. This suggests that the f orbitals of each R3+ (R = Tb, Dy) ion are engaged, though weakly, in bonding with the surrounding ligand atoms. The magnetic properties of GdInO3 with the zero orbital moment are adequately described by the spin exchanges extracted by DFT + U calculations. In describing the magnetic properties of TbInO3 and DyInO3 with nonzero orbital moments, however, the spin exchanges extracted by DFT + U + SOC calculations are necessary. The spin exchanges of RInO3 (R = Gd, Tb, Dy) are dominated by the two NN spin exchanges J(1) and J(2) of their hexagonal spin lattice, in which the honeycomb lattice of J(2) forms spin-frustrated (J(1), J(1), J(2)) triangles. The J(2)/J(1) ratios are calculated to be similar to 3, similar to 1.7, and similar to 1 for GdInO3, TbInO3, and DyInO3, respectively. This suggests that the antiferromagnetic (AFM) ordering of GdInO3 below 1.8 K is most likely an AFM ordering of its honeycomb spin lattice and that TbInO3 would exhibit low-temperature magnetic properties similar to those of GdInO3 while DyInO3 would not.