This paper introduces a new evolving fuzzy-rule-based algorithm for online data streams, named Extreme Value evolving Predictor (EVeP). It offers a statistically well-founded approach to defining the evolving fuzzy granules that form the antecedent and the consequent parts of the rules. The evolving fuzzy granules correspond to radial inclusion Weibull functions. They are interpreted by the Extreme Value Theory as the limiting distribution of the relative proximity among the rules of the learning model. Regarding the parameters of the Takagi-Sugeno term at the consequent of the rules, the algorithm enhances the already demonstrated benefits of Multitask Learning by replacing a binary version with a fuzzy structural relationship among the rules. The pairwise similarity among the rules is automatically provided by the current interaction of the evolving fuzzy granules at the antecedent and at the consequent parts of their corresponding rules. Several computational experiments, using artificial and real-world time series, attest to the dominating prediction performance of EVeP when compared to state-of-the-art evolving algorithms.