Rectangular graphene nanodots have both armchair and zigzag edges, which can be terminated with a variety of atoms or molecular groups. Our first-principles study shows that edge chemical modification can alter the electronic structure of graphene nanodots significantly. We find that when saturated with different atoms or molecular groups (such as -H, -F, -OH, etc.) on the zigzag edges, the graphene nanodots show a spin-polarized ground state, but with magnetic moment, spin density and energy gap are strongly dependent on the type of saturating atom or molecular group. Our results indicate that graphene nanodots have great potential to serve as future molecular sensor and transistor devices.