Bayesian optimization (BO) is increasingly used in molecular optimization and to guide self-driving laboratories for automated materials discovery. A crucial aspect of BO is how molecules and materials are represented as feature vectors, where both the completeness and compactness of these representations can influence the efficiency of the optimization process. Traditionally, a fixed representation is chosen by expert chemists or applying data-driven feature selection methods on available labelled datasets. However, when dealing with novel optimization tasks, prior knowledge or large datasets are often unavailable, and relying on these even can introduce bias into the search process. In this work, we demonstrate a Feature Adaptive Bayesian Optimization (FABO) framework, which integrates feature selection in Bayesian optimization process to dynamically adapt material representations throughout the optimization cycles. We demonstrate the effectiveness of this adaptive approach across several molecular optimization tasks, including the discovery of high-performing metal-organic frameworks (MOFs) in three distinct tasks, each involving unique property distributions and requiring a distinct representation. Our results show that the adaptive nature of the representation leads to outperforming random search baseline and scenarios where prior knowledge of the feature space is available. Notably, for known optimization tasks, FABO automatically identifies representations that are aligned with human chemical intuition, validating its utility for optimization tasks where such insights are not available in advance. Lastly, we show how a biased representation can adversely impact BO performance, highlighting the importance of adaptive representation to different tasks. Our findings highlight FABO as a robust approach for navigating large, complex materials search spaces in automated discovery campaigns.