Despite extensive studies in the past, deterioration of mechanical properties due to hydrogen environment exposure remains a serious problem for structural materials. More effective improvement of a material's resilience requires advanced computational methods to elucidate the fundamental mechanisms of the hydrogen effects. To enable accurate molecular dynamics (MD) studies of the hydrogen effects on metals, we have developed a high-fidelity analytical bond-order potential (BOP) for the copper-hydrogen binary system as a representative case. This potential is available through the publically available MD code LAMMPS. The potential parameters are optimized using an iterative process. First, the potential is fitted to static and reactive properties of a variety of elemental and binary configurations including small clusters and bulk lattices (with coordination varying from 1 to 12). Then the potential is put through a series of rigorous MD simulation tests (e.g., vapor deposition and solidification) that involve chaotic initial configurations. It is demonstrated that this Cu-H BOP not only gives structural and property trends close to those seen in experiments and quantum mechanical calculations, but also predicts the correct phase transformations and chemical reactions in direct MD simulations. The correct structural evolution from chaotic initial states strongly verifies the transferability of the potential. A highly transferable potential is the reason that a well-parameterized analytical BOP can enable MD simulations of metal-hydrogen interactions to reach a fidelity level not achieved in the past.