It was recently argued by Hooper and Goodenough that the excess gamma-ray emission from within 1°-2°of the Galactic center can be well described by annihilation of ∼8GeV dark matter particles into tau pairs. I show that such a dark matter signal can be obtained naturally in the lepton-specific two-Higgs-doublet model extended by a stable singlet scalar dark matter candidate. The favored parameter region prefers a light Higgs state (below 200 GeV) with enhanced couplings to leptons and a sizable invisible branching fraction. Part of the favored region leads to invisible decays of both of the CP-even neutral Higgs states.