Browser fingerprinting is a tracking technique that collects attributes and calls functions from the browser's APIs. Unlike cookies, browser fingerprints are difficult to evade or delete, raising significant privacy concerns for users as they can be used to re-identify individuals over browsing sessions without their consent. Yet, there has been limited research on the impact of browser configuration settings on these fingerprints. This paper introduces FP-Rainbow, a novel approach to systematically explore and map the configuration space of Chromium-based web browsers aiming to identify the impact of configuration parameters on browser fingerprints and their changes over time. We explore 1,748 configuration parameters (switches) and identify their impact on the browser's BOM (Browser Object Model). By collecting and analyzing over 61,000 fingerprints from 18 versions of Chromium, our study reveals that 32 to 56 of these configuration parameters (depending on versions), such as disable-3d-apis or disable-notifications, influence the fingerprint of a web browser. FP-Rainbow also proves efficient in identifying browser configuration parameters from unknown fingerprints, achieving an average successful identification rate of 84% when considering a single configuration parameter and 78% when multiple parameters are involved, across all evaluated browser versions. These findings emphasize the importance of measuring the impact of configuration parameters on browsers to develop safer and more privacy-friendly web browsers.