We report the triton (t) production in midrapidity (jyj < 0.5) Au + Au collisions at ffiffiffiffiffiffiffip= sNN 7.7-200 GeV measured by the STAR experiment from the first phase of the beam energy scan at the Relativistic Heavy Ion Collider. The nuclear compound yield ratio (Nt x Np=N2d), which is predicted to be sensitive to the fluctuation of local neutron density, is observed to decrease monotonically with increasing charged-particle multiplicity (dNch=d eta) and follows a scaling behavior. The dNch=d eta dependence of the yield ratio is compared to calculations from coalescence and thermal models. Enhancements in the yield ratios relative to the coalescence baseline are observed in the 0%-10% most central collisions at 19.6 and 27 GeV, with a significance of 2.3 sigma and 3.4 sigma, respectively, giving a combined significance of 4.1 sigma. The enhancements are not observed in peripheral collisions or model calculations without critical fluctuation, and decreases with a smaller pT acceptance. The physics implications of these results on the QCD phase structure and the production mechanism of light nuclei in heavy-ion collisions are discussed.