Fluid antenna is a new reconfigurable antenna technology that can dynamically adjust the positions or ports of radiating elements and therefore provides a new degree of freedom for wireless communications. However, the associated port selection is a challenging large-scale combinatorial optimization problem and difficult to solve. Existing manually designed heuristic algorithms are not only labor-intensive, but cannot achieve satisfactory performance. In this paper, we propose a novel paradigm that leverages large language models (LLMs) for automated design of optimization algorithms for fluid antenna systems without manual hyperheuristic tuning. Specifically, we study the problem of maximizing the minimum signal-to-interference-plus-noise ratio (SINR) in the downlink to ensure fairness among users by optimizing port selection and beamforming. We investigate two LLM-enabled algorithm optimization strategies. The first is to optimize the crossover and mutation operations to enhance the performance of the well-known genetic algorithm and the second is to design AutoPort, a new heuristic from scratch by LLM, to solve the optimization problem. Simulation results verify that the proposed method can achieve near-optimal performance and significant improvement over the conventional genetic algorithm and the deep learning approach.