Geothermal energy is utilized due to its sustainability and ability to provide a continuous low-emission source of heat, making it an ideal complement to the high-efficiency requirements of modern power plants. This study investigates a novel configuration coupling a solid oxide fuel cell-gas turbine unit with a dual-flash binary geothermal structure, enhanced by low-temperature electricity generation and hydrogen production subsystems. This configuration incorporates a regenerative steam Rankine cycle and a proton exchange membrane electrolyzer to achieve an efficient overall design. The setup is evaluated from thermodynamic and economic perspectives using a non-dominated sorting genetic algorithm-II for optimization. A complete parametric study assesses the sensitivity of critical variables. Multi-objective optimization is conducted across three scenarios considering exergy efficiency, hydrogen production rate, sum unit cost of products, and the payback period as objective functions. Results indicate an optimal exergy efficiency of 56.95% and a hydrogen production rate of 0.22 kg/h, with a product unit cost of $7.06/GJ and a payback period of 1.12 years. The parametric study underscores the significant impact of the number of solid oxide fuel cells and their existing density on key variables of the presented configuration. This study highlights the system's potential for efficient and economically feasible power and hydrogen production.