Weak decays of the vector D ∗ 0 and B ( s ) ∗ 0 mesons to the μ + μ - final state provide novel potential to test the Standard Model of particle physics. Such processes have extremely small branching fractions as the vector mesons are able to decay through electromagnetic and (for the D ∗ 0 meson) strong interactions. Nonetheless, the production of copious quantities of these particles in LHC collisions, and the ability to exploit experimental techniques that can suppress background to low levels, provides good potential to reach interesting sensitivity. The possibility to reconstruct these processes as part of the decay chain of B - or B c + mesons appears particularly attractive due to the clean experimental signature of the displaced vertex. Indeed, published LHCb data on B - → π - μ + μ - decays already implies a stringent limit on the branching fraction of D ∗ 0 → μ + μ - . Estimates are made on the achievable sensitivity to D ∗ 0 → μ + μ - and B ( s ) ∗ 0 → μ + μ - decays with the LHCb experiment.