We perform density functional calculations to investigate the adsorption of molecular hydrogen on Ti-doped zigzag graphene nanoribbons using a nonlocal van der Waals functional that has recently been proposed for accurate description of exchange and correlation effects in weakly bound systems. Our results show that the adsorption of a single H2 molecule is dissociative in purely energetic terms, but there exists an energy barrier that prevents dissociation when the molecule is deposited on the Ti-doped graphene nanoribbon. When the Ti atom is adsorbed at a central or lateral hole site, each atom can bind up to four H2 molecules, in each case satisfying the binding energy criterion specified by the U.S. Department of Energy for novel hydrogen-storage materials. On this basis, one can consider an effective hydrogen coverage on Ti-coated graphene nanoribbons with gravimetric density beyond the target of 6 %.