We have performed ab initio density functional theory calculations, incorporating London dispersion corrections, to study the absorption of molecular hydrogen on zigzag graphene nanoribbons whose edges have been functionalized by OH, NH2, COOH, NO2, or H2PO3. We find that hydrogen molecules always preferentially bind at or near the functionalized edge, and display induced dipole moments. Binding is generally enhanced by the presence of polar functional groups. The largest gains are observed for groups with oxygen lone pairs that can facilitate local charge reorganization, with the biggest single enhancement in adsorption energy found for "strong functionalization" by H2PO3 (115 meV/H2 versus 52 meV/H2 on bare graphene). We show that for binding on the "outer edge" near the functional group, the presence of the group can introduce appreciable contributions from Debye interactions and higher-order multipole electrostatic terms, in addition to the dominant London dispersion interactions. For those functional groups that contain the OH moiety, the adsorption energy is linearly proportional to the number of lone pairs on oxygen atoms. Mixed functionalization with two different functional groups on a graphene edge can also have a synergistic effect, particularly when electron-donating and electron-withdrawing groups are combined. For binding on the "inner edge" somewhat farther from the functional group, most of the binding again arises from London interactions; however, there is also significant charge redistribution in the π manifold, which directly reflects the electron donating or withdrawing capacity of the functional group. Our results offer insight into the specific origins of weak binding of gas molecules on graphene, and suggest that edge functionalization could perhaps be used in combination with other strategies to increase the uptake of hydrogen in graphene. They also have relevance for the storage of hydrogen in porous carbon materials, such as activated carbons.