This paper presents BEASST (Behavioral Entropic Gradient-based Adaptive Source Seeking for Mobile Robots), a novel framework for robotic source seeking in complex, unknown environments. Our approach enables mobile robots to efficiently balance exploration and exploitation by modeling normalized signal strength as a surrogate probability of source location. Building on Behavioral Entropy(BE) with Prelec's probability weighting function, we define an objective function that adapts robot behavior from risk-averse to risk-seeking based on signal reliability and mission urgency. The framework provides theoretical convergence guarantees under unimodal signal assumptions and practical stability under bounded disturbances. Experimental validation across DARPA SubT and multi-room scenarios demonstrates that BEASST consistently outperforms state-of-the-art methods, achieving 15% reduction in path length and 20% faster source localization through intelligent uncertainty-driven navigation that dynamically transitions between aggressive pursuit and cautious exploration.