The recently introduced concept of synthetic dimensions allows for the realization of higher-dimensional topological phenomena in lower-dimensional systems. In this paper, we propose a setup where synthetic dimensions arise in mesoscopic hybrid devices and discuss how they provide a natural route to topological states. We demonstrate this for the current induced into a closed one-dimensional Aharonov-Bohm ring by the interaction with a dynamic mesoscopic magnet. The quantization of the magnetic moment provides a synthetic dimension that complements the charge motion around the ring. We present a direct mapping that places the combined ring-magnet system into the class of quantum Hall models and demonstrate that topological features, combined with the magnet's anisotropy, can lead to clear signatures in the persistent current of the single-particle ground state. Our synthetic-dimension model also extends to the many-electron case, where the collective electronic motion couples with the magnet.