The present study introduces a novel hydrogen liquefaction system that integrates a Claude cycle and a branched GAX cycle, marking the first instance of such a combination. Utilizing solar energy improves hydrogen manufacturing efficiency and sustainability. This research uses actual data to estimate the best operating time by examining thermodynamic performance, payback duration, exergoeconomic parameters, environmental effect, and sustainability. A current bi-objective optimization technique maximizes exergy efficiency and minimizes hydrogen liquefaction cost. Evaporator and generator temperatures, solar direct beam irradiation, and cycle's high pressure are key decision variables. The sensitivity analysis highlights the substantial influence of cycle's high pressure on hydrogen liquefaction cost, as indicated by a sensitivity index of 0.486. The research calculates the best solution using TOPSIS decision-making, resulting in 50.22 % exergy efficiency and $1.366/kg hydrogen liquefaction cost. After conducting a thorough case study, it becomes evident that May is the most optimal month for liquid hydrogen production, payback period, and net profit. However, January has a high coefficient of performance and exergy efficiency and low liquid hydrogen production cost. The cheapest capital investment month is July. This complete study of the solar-driven hydrogen liquefaction system uncovers critical parameters and their effects, enabling hydrogen production efficiency and sustainability.