On the basis of first-principles calculations of molecular electron structure, we discuss the strategy of modifying the carbon-based materials in order to increase their capacity to bind with molecular hydrogen. In particular, we have studied hydrogen adsorption on molecular complexes having anionic aromatic carbon-based rings stabilized by cations of alkali (Li⁺, Na⁺, K⁺) or alkali-earth metals (Be²⁺, Mg²⁺, Ca²⁺). The adsorption depends more on the properties of the cation than on the ring itself. The interaction of the H₂ molecule with an electrostatic field leads to the binding of the hydrogen molecule with the strongly polarized ionic molecular complex. The number of the adsorbed molecules is driven by two factors acting in opposite directions: the binding energy, which should be larger than a 4–5 kJ/mol threshold needed to keep hydrogen molecules attached, and the area around the cation (coordination sphere), which is determined by its radius. As a compromise between these factors, we propose several promising candidates for building blocks of hydrogen storage materials, including diboratabenzene lithium, C₄B₂H₆Li₂, and diboratabenzene potassium, C₄B₂H₆K₂, which can adsorb 6 and 12 H₂ molecules, correspondingly. We also discuss the possibility of linking these molecular complexes in three-dimensional structures.