Carbon materials, such as graphene nanoflakes, carbon nanotubes, and fullerene, can be widely used to store hydrogen, and doping these materials with lithium (Li) generally increases their H2-storage densities. Unfortunately, Li is expensive; therefore, alternative metals are required to realize a hydrogen-based society. Sodium (Na) is an inexpensive element with chemical properties that are similar to those of lithium. In this study, we used density functional theory to systematically investigate how hydrogen molecules interact with Na-doped graphene nanoflakes. A graphene nanoflake (GR) was modeled by a large polycyclic aromatic hydrocarbon composed of 37 benzene rings, with GR-Na-(H2)n and GR-Na+-(H2)n (n = 0–12) clusters used as hydrogen storage systems. Data obtained for the Na system were compared with those of the Li system. The single-H2 GR-Li and GR-Na systems (n = 1) exhibited binding energies (per H2 molecule) of 3.83 and 2.72 kcal/mol, respectively, revealing that the Li system has a high hydrogen-storage ability. This relationship is reversed from n = 4 onwards; the Na systems exhibited larger or similar binding energies for n = 4–12 than the Li-systems. The present study strongly suggests that Na can be used as an alternative metal to Li in H2-storage applications. The H2-storage mechanism in the Na system is also discussed based on the calculated results.