Biological and living systems process information across spatiotemporal scales, exhibiting the hallmark ability to constantly modulate their behavior to ever-changing and complex environments. In the presence of repeated stimuli, a distinctive response is the progressive reduction of the activity at both sensory and molecular levels, known as habituation. In this work, we solve a minimal microscopic model devoid of biological details, where habituation to an external signal is driven by negative feedback provided by a slow storage mechanism. We show that our model recapitulates the main features of habituation, such as spontaneous recovery, potentiation, subliminal accumulation, and input sensitivity. Crucially, our approach enables a complete characterization of the stochastic dynamics, allowing us to compute how much information the system encodes on the input signal. We find that an intermediate level of habituation is associated with a steep increase in information. In particular, we are able to characterize this region of maximal information gain in terms of an optimal trade-off between information and energy consumption. We test our dynamical predictions against experimentally recorded neural responses in a zebrafish larva subjected to repeated looming stimulations, showing that our model captures the main components of the observed neural habituation. Our work makes a fundamental step towards uncovering the functional mechanisms that shape habituation in biological systems from an information-theoretic and thermodynamic perspective.