Estimating the costs and performance of emerging separation technologies is technically challenging because of the complex design space and the interdependence between system and module-level design decisions. This work develops a process-based cost-optimization model that minimizes the levelized cost of water (LCOW) produced using multi-stage gap MD. The model fully captures solution properties, heat transport, and mass transport relationships within each membrane module, specifies the optimal multi-stage configuration of modules in series, and determines cost-optimal operation of each stage and the system as a whole as a function of gap type, the number of stages, feed conditions, and desired water recovery. We estimate that the LCOW for recovering 50% of a 100 g/L feed using conductive gap multi-stage MD is approximately 6.5 $ per m3 of product water. Cost-optimal designs of multi-stage systems balance operational costs (primarily energy) and capital costs (primarily membrane). This balance results in stages having a different design and operation across the system (e.g. initial stages operating with chilling and heating and end stages using only heating). Sensitivity analysis on the model results suggests that improving membrane parameters and module designs may provide modest cost reductions of <15%.