Automated Machine Learning (AutoML) has revolutionized the development of machine learning pipelines. However, its application to data streams presents unique challenges. While significant progress has been made in streaming classification, advancements in streaming regression remain limited. To address this gap, we propose Auto-Reg, an AutoML framework designed specifically for data stream regression. Auto-Reg introduces two key components: a dynamic budget adjustment mechanism for efficient resource allocation and a Probability-Weighted Hyperparameter Search (PWHS) strategy that balances exploration and exploitation. Comprehensive experiments on both real-world and synthetic datasets, supported by theoretical and empirical evaluations, demonstrate that Auto-Reg consistently outperforms state-of-the-art data stream regression models in terms of predictive accuracy.