Conventional stochastic response surface method (SRSM) based on polynomial chaos expansion (PCE) for uncertainty propagation treats every sample points equally during the regression process and may produce inaccurate coefficient estimations in PCE. A new weighted stochastic response surface method (WSRSM) to overcome such limitation by considering the sample probabilistic weights in regression is studied in this work. Techniques that associate sample probabilistic weights to different sampling approaches such as Gaussian Quadrature point (GQ), Monomial Cubature Rule (MCR) and Latin Hypercube Design (LHD) are developed. The proposed method is demonstrated by several mathematical and engineering examples. Results show that for various sampling techniques, WSRSM can consistently improve the accuracy of uncertainty propagation compared to the conventional SRSM without adding extra computational cost. Insights into the relative accuracy and efficiency of using various sampling techniques in implementation are provided.