Using the high-resolution Q3D magnetic spectrograph at the Maier-Leibnitz Laboratory (MLL) Tandem accelerator in Munich, we studied 0+ excitations in the mercury isotopes 198Hg, 200Hg, and 202Hg after two-neutron transfer. We only observed 4-6 excited 0+ states per nucleus up to about 3-MeV excitation energy, far fewer than in other experiments of this (p, t) campaign. The results reveal a sharp drop in the number of low-lying 0+ states towards the 208Pb shell closure. We discuss the low-energy 0+ state density as a function of the valence nucleon number Nval. The 0+ excitation energies and the measured (p, t) transfer cross sections indicate a structural change throughout the Hg isotopes, with the most notable result being the peaking in the cross section of the low-lying excited 02+ state in 200Hg.