No peer-reviewed study has tested thunderstorm sounds specifically for cognition or ADHD. Thunderstorm is a compound soundscape that combines continuous rain (strong general evidence via Buxton et al. 2021 PNAS meta-analysis) with sporadic low-frequency thunder rumble.
The continuous rain layer provides broadband masking and the steady-state arousal benefit associated with water sounds. The distant thunder rumble (predominantly below 200 Hz) provides a sense of "being away," one of the four components of Attention Restoration Theory (Kaplan, 1995). The combination of continuous and punctuated elements creates a sound environment that is simultaneously steady and unpredictable, preventing habituation.
Included based on the evidence for its component parts, not direct testing of the combined soundscape.