The geo-storage of hydrogen (H2) is essential for advancing a robust and industrial-scale H2-based economy. The efficient H2 storage in a geological formation depends on the presence of a secure and impermeable caprock, such as a shale formation. However, the existing literature lacks information on the kinetics of H2 adsorption in actual shale formations with diverse organic contents and mineral compositions. Therefore, the kinetics of H2 adsorption in organic-rich shale samples from a Jordanian oil (JO) source rock formation are experimentally investigated herein at various temperatures (193, 273, 303, and 333 K) and two equilibrium pressures (15 and 45 bar) by using a volumetric method. The impact of mineral compositions and total organic content (TOC) on H2 adsorption is studied using two samples with distinct mineralogy and TOC values. Thermodynamic calculations are performed, and common adsorption models are used to analyze the experimental data. The results indicate that the H2 adsorption rate increases with pressure and decreases with temperature, being characterized by an initial rapid adsorption phase followed by a slower adsorption period as the uptake of H2 approaches equilibrium. More interestingly, the rate of H2 adsorption is higher in the sample with a high TOC and a richer composition of carbonate minerals. Both the applied adsorption isotherm models and adsorption kinetics models fit the experimental data remarkably well, thereby indicating their suitability for analyzing the kinetics of H2 adsorption in the JO shale. The unipore model matches the H2 kinetics data well, with R2 > 0.91. The H2 diffusion coefficient increases with pressure and temperature, where the H2 diffusion in shale accelerates with thermal motion intensity. This study provides fundamental insights into the adsorption kinetics of H2 and is expected to assist in implementing an industrial H2-based economy.