Hydrogen-based compounds under ultrahigh pressure, such as the polyhydrides H3S and LaH10, superconduct through the conventional electron-phonon coupling mechanism to attain the record critical temperatures known to date. Here we exploit the intrinsic advantages of hydrogen to strongly enhance phonon-mediated superconductivity in a completely different system, namely, a two-dimensional material with hydrogen adatoms. We find that van Hove singularities in the electronic structure, originating from atomiclike hydrogen states, lead to a strong increase of the electronic density of states at the Fermi level, and thus of the electron-phonon coupling. Additionally, the emergence of high-frequency hydrogen-related phonon modes in this system boosts the electron-phonon coupling further. As a concrete example, we demonstrate the effect of hydrogen adatoms on the superconducting properties of monolayer MgB2, by solving the fully anisotropic Eliashberg equations, in conjunction with a first-principles description of the electronic and vibrational states, and their coupling. We show that hydrogenation leads to a high critical temperature of 67 K, which can be boosted to over 100 K by biaxial tensile strain.