One potential solution for reducing carbon dioxide emissions from ships and meeting the Energy Efficiency Existing Ship Index (EEXI) requirements is to use a hybrid propulsion system that combines liquid hydrogen and liquefied natural gas fuels. To improve energy efficiency for diesel-electric dual-fuel ship propulsion systems, an engine power limitation system can also be used. This paper examines the potential use of these systems with regard to several factors, including compliance with EEXI standards set by the International Maritime Organization, fuel ratio optimisation, installation requirements, and economic feasibility. As a case study, an LNG carrier is analysed, with dual-fuel diesel-electric and two hybrid systems adjusted to meet IMO-EEXI requirements with engine power limitation percentages of 25%, 0% (hybrid option 1), and 15% (hybrid option 2), respectively. From an economic standpoint, the liquid hydrogen-based system has competitive costs compared to the dual-fuel diesel-electric system, with costs of 2.1 and 2.5 dollars per kilogram for hybrid system options 1 and 2, respectively.