We investigate η-Ricci-Bourguignon solitons on hyperbolic spaces Hn, with a focus on the Poincaré half-plane H2. Building upon recent advances on classical Ricci-Bourguignon solitons in this geometric setting, we establish necessary and sufficient conditions for the existence of such structures and analyze the properties of the underlying potential vector fields. The results obtained on H2 are subsequently extended to higher-dimensional hyperbolic spaces.