The doping dependence of specific heat and magnetic susceptibility has been investigated for Li(x)ZrNCl superconductors derived from a band insulator. As the carrier concentration is decreased, the anisotropy of superconducting gap changes from highly anisotropic to almost isotropic. It was also found that, upon reducing carrier density, the superconducting coupling strength and the magnetic susceptibility are concomitantly enhanced in parallel with T(c), while the density of states at the Fermi level is kept almost constant. Theoretical calculations taking into account the on-site Coulomb interaction reproduced the experimental results, suggesting a possible pairing mediated by magnetic fluctuations, even in the doped band insulators.