The dipole moment, parallel dipole polarizability, and the parallel first hyperpolarizability of the coinage metal hydrides (CuH, AgH, AuH) are studied in the context of their vibrational contributions. The partition of the property values into pure electronic, vibrational averaging, and pure vibrational terms is discussed, and the corresponding contributions are calculated by means of the field-dependent Numerov-Cooley integration technique. The potential energy and property surfaces, required for the evaluation of these contributions, follow from different high-level electronic structure calculations. Parallel to the nonrelativistic studies, a relativistic treatment of the investigated properties has been carried out. It has been found that relativistic effects significantly reduce the magnitude of the vibrational corrections to the studied properties. A particularly large relativistic reduction has been found for the vibrational contribution to the first hyperpolatizability. The relativistic effect is enhanced with increasing nuclear charge of the heavy atom, and can be interpreted in terms of the relativistic effect on the potential energy and electric property curves.