The homotopy stochastic residual error method (HSREM) is proposed to address the structural elastic stability problem involving large fluctuation in random parameters, and the statistics of the buckling load and buckling mode are obtained. In HSREM, the buckling load and buckling mode of the structure are expressed using the homotopy series, and the explicit recursive expression of the coefficients in the homotopy series are derived through the homotopy analysis method. Further, the stochastic residual error is defined based on substituting the expressions of the buckling load and buckling mode into the buckling governing equation. By means of minimizing the stochastic residual error, the optimal form of the homotopy series is determined automatically, which overcomes the drawbacks of the exist homotopy stochastic finite element method that the computational accuracy affected by the selected samples and relying on empirical knowledge. In addition, for a structure involving large fluctuation in random parameters, the proposed method has remarkable ability in convergence and stability when the higher-order terms in the homotopy series are employed, while results from the traditional perturbation method based on the Taylor series may divergent. And HSREM has a computational efficiency compared with the Monte Carlo simulation method. The validity of the proposed method is verified through the stability analysis of a variable cross-section column subjected to axial force and a 7-story frame structure.