  The origin of trace hydrogen in white dwarfs (WDs) with He-dominated
atmospheres is a long-standing problem, one that cannot satisfactorily be
explained by the historically-favoured hypothesis of accretion from the
interstellar medium. Here we explore the possibility that the gradual accretion
of exo-Oort cloud comets, which are a rich source of H, contributes to the
apparent increase of trace H with WD cooling age. We determine how often
remnant exo-Oort clouds, freshly excited from post-main-sequence stellar mass
loss, dynamically inject comets inside the WD's Roche radius. We improve upon
previous studies by considering a representative range of single WD masses
(0.52-1.00 Solar masses) and incorporating different cloud architectures, giant
branch stellar mass loss, stellar flybys, Galactic tides and a realistic escape
ellipsoid in self-consistent numerical simulations that integrate beyond 8 Gyr
ages of WD cooling. We find that about 10^{-5} of the material in an exo-Oort
cloud is typically amassed onto the WD, and that the H deposits accumulate even
as the cloud dissipates. This accumulation may account for the relatively large
amount of trace H, 10^{22}-10^{25} g, that is determined frequently among WDs
with cooling ages >= 1 Gyr. Our results also reaffirm the notion that exo-Oort
cloud comets are not the primary agents of the metal budgets observed in
polluted WD atmospheres.
