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The search for optimal phase change materials (PCMs) for latent heat thermal energy storage systems (LHTESS) focuses almost exclusively on the properties of the PCM. This neglects the significant contribution of the cost of the containment vessel on the total cost of the LHTESS. Thus, to accurately assess the thermoeconomic performance of various PCMs, the relationship between the cost of the containment vessel and the properties of the PCM must be understood. This paper presents an analytical method for optimizing the design of tube and shell or tube and fin containment vessels for the least cost, subject to geometric and performance constraints. One of the key performance constraints is the time to charge, whose evaluation typically requires computationally-expensive simulations which are unsuitable for optimization algorithms. To enable efficient optimization, a novel closed-form approximation of the charge time is developed and validated through numerical simulations. Subsequently, the optimization methodology is used to investigate the relationship between the optimal vessel geometry and configuration, fin properties, PCM properties, and cost of two PCMs. For these PCMs, the use of high thermal conductivity fins is shown to dramatically reduce the total cost of the system. The new methodology is an efficient way to compare the thermoeconomic performance of PCMs as latent heat storage solutions, allowing for the accurate assessment of PCMs. Keywords PCM heat exchanger design; Latent heat thermal energy storage system optimization; Phase change material; Thermal energy storage