Ubiquitously found in the Universe, atomic hydrogen represents up to 70% of the neutral gas composition of the Milky Way. As an adatom, hydrogen can physisorb or chemisorb onto interstellar dust grains and icy mantles, thereby contributing to the formation of H2 and, potentially, to the synthesis of more complex hydrogenated species. In addition, structures of relatively large specific surface areas -- such as silicates, amorphous carbon, graphene sheets, or water ice-host heterogeneous chemistry that is thought to facilitate the emergence of complex organic matter in astrophysical environments. Although the fundamental physical and chemical processes occurring at dust/gas interfaces are well characterized, current understanding of dust properties governing the formation of H2 and complex molecules remains incomplete. In this context, we introduce graphitic-like two-dimensional carbon nitride monolayer structures (2D-CN) as a putative molecular family of potential relevance to astrochemistry. The physicochemical and electronic properties of these materials have been extensively examined in recent years for industrial and technological applications. Here, we propose that their importance may likewise extend to interstellar and circumstellar environments. To explore this possibility, we employed Density Functional Theory (DFT) calculations to investigate the characteristics and extent of H adsorption onto C2N1, C3N1, C3N2, C3N4, C4N3, C6N6, C6N8, C9N4, and C9N7 monolayer nanosheets. We identify multiple adsorption sites over C-C bonds, above C and N atoms, and hollow (macropore) locations at which energetically favorable binding of atomic hydrogen could occur in the interstellar medium (ISM). From an astrochemical perspective, these 2D-CN structures, if formed, could therefore contribute to the physicochemical processing and evolution of hydrogen in the ISM.