In a recent paper, four of the present authors proposed a class of dark matter models where generalized parity symmetry leads to equality of dark matter abundance with baryon asymmetry of the Universe and predicts dark matter mass to be around 5 GeV. In this paper, we explore how this model can be tested in direct search experiments. In particular, we point out that if the dark matter happens to be the mirror neutron, the direct detection cross section has the unique feature that it increases at low recoil energy unlike the case of conventional weakly interacting massive particles. It is also interesting to note that the predicted spin-dependent scattering could make significant contribution to the total direct detection rate, especially for light nucleus. With this scenario, one could explain recent DAMA and CoGeNT results.