A new method for dynamic sampling of Kriging surrogate models for uncertainty quantification is developed and presented. The criterion for the dynamic adaptive sampling proposed is based on combining the expected uncertainty of the fit and the gradient information resulting from the Kriging predictors, and an error-estimate term (based on the difference in the Kriging predictors with different correlation length scales). The Kriging-based dynamic adaptive sampling method proposed is tested on two-dimensional analytic functions with smoothly and steeply varied responses in the quantities of interest under normal uncertainty distributions. Compared with a classical polynomial chaos expansion method based on the Gauss quadrature rule and a dynamic adaptive sampling method based only on the uncertainty of the Kriging predictor fit, this new method shows superior performance for estimating the statistics of the quantity of interest in terms of both accuracy and robustness, and regardless of either the choice of the initial set of samples or the smoothness of the stochastic space.