By using a configuration interaction approach with up to the fifth excitations taken into account, we study the excitonic effect in the optical absorption in graphene nanodots. While the many-electron states are either singlet or doublet in a triangular nanodot system, all the excited singlet states are found to be optically dark in the absorption. These dark excitons are shown to originate mainly from the geometric symmetry of the system and would remain inactive even when the electron-hole or sublattice symmetry is broken. The first excited state in most of the cases is found to be a dark singlet; however, the order of dark and bright excitonic states is shown to be quite sensitive to the strength of electron-electron interactions such as the dielectric screening from the substrate. All the double degeneracies in the excitonic spectrum are found to be lifted when the rotational symmetry is absent such as in the case of a trapezoidal nanodot; however, the first excited state is shown to still remain a dark exciton when there is a strong screening effect. In order that the optical gap of a graphene nanodot can be efficiently tuned by its dielectric environment, the geometric symmetry is revealed to be a crucial factor.