The (perhaps implicit) assumption of the device' design could be that every thunderstorm is a cold front underpassing wedge like a layer of ground level warm and humid air. Then, while surface / ground air temperature drops, the temperature of the exposed device equally decreases. Often, lower temperature of a solvent corresponds to lower solvent capacity for the solute (keyword solubility product) and precipitation to establish (again) a new equilibrium between material as solid and dissolved material. This could lead to a less transparent solution in the tube recognized by the trained eye if the device is constantly in an instrument shelter.
Further on, while the humid warm air is pushed upwards by the cold front, these masses' temperature decreases because they reach higher heights. Then, their capacity for gaseous water is reduced (keyword humidity). This leads to condensation to form little droplets of water (related: Mollier steam diagrams); depending on currents of rising air (updraft) the potential for rain, and eventually thunderstorms. For such a mechanism leading to a thunderstorm, the concept could work, possibly complemented with visual observation (slow formation of characteristic shapes of clouds, etc). In comparison to an electronic device, it wouldn't be easy to be used while hiking in alpine regions, where swifter changes of weather merit a likely faster response to changes (time constant in signal processing).
But recording the temperature (variation) as correlation to and sufficient predictor of thunderstorms isn't sufficient. Assume you placed the device in your living room (like some do for an aneroid barometer) for convenience of reading. It is plausible that if you air your room in winter, the mere drop of temperature the device is exposed to now leads to a reading "storm ahead". A better approach to predict the likelihood of a storm includes monitoring air pressure (and its variation) both over time, as well as in space. Literally, there are ground (or surface / sea level) highs and lows of air pressure, and in greater heights altitude highs and lows. Currents of air / storms as a release of pressure gradients then may be vertical, horizontal (e.g. updrift as used in gliding), attenuated by woods, and in more complex trajectories because of obstacles like mountains in the terrain. It is not obvious how this setup is sensible to these variations.