Selecting the most suitable renewable energy technology among feasible alternatives considering conflicting criteria is a Multiple Criteria Decision Making (MCDM) problem. One of the essential stages in the methods used to solve such problems is determining the appropriate weight of each criterion to be considered. The Shannon Entropy method is a frequently used MCDM method to calculate the criteria weights, however it is not suitable to solve problems for which uncertainty in the input data exists. This paper presents a new extended Shannon Entropy method: the Integrated Constrained Fuzzy Shannon Entropy (IC-FSE) method, by which criteria weights are obtained from uncertain input data. To show the applicability of IC-FSE, an illustrative example for the selection of a renewable energy technology in the mining industry is presented, in which three alternative renewable energy technologies, onshore wind, solar photovoltaic and concentrated solar power, were evaluated with respect to technical, social, economic and environmental categories. The results show that IC-FSE can effectively provide appropriate fuzzy solutions for weighting the sustainability criteria for renewable energy technologies. The superiority of this method is showcased by demonstrating that IC-FSE results are more robust than those obtained using other existing methods. The methodology presented can be applied broadly in the renewable energy sector to ensure better informed decision making processes.