We report diffuse reflectivity measurements in InNbO4, ScNbO4, YNbO4, and eight rare-earth niobates. A comparison with established values of the bandgap of InNbO4 and ScNbO4 shows that Tauc plot analysis gives erroneous estimates of the bandgap energy. Conversely, accurate results are obtained considering excitonic contributions using the Elliot-Toyozawa model. The bandgaps are 3.25 eV for CeNbO4, 4.35 eV for LaNbO4, 4.5 eV for YNbO4, and 4.73-4.93 eV for SmNbO4, EuNbO4, GdNbO4, DyNbO4, HoNbO4, and YbNbO4. The fact that the bandgap energy is affected little by the rare-earth substitution from SmNbO4 to YbNbO4 and the fact that they have the largest bandgap are a consequence of the fact that the band structure near the Fermi level originates mainly from Nb 4d and O 2p orbitals. YNbO4, CeVO4, and LaNbO4 have smaller bandgaps because of the contribution from rare-earth atom 4d, 5d, or 4f orbitals to the states near the Fermi level.