Energy storage is a crucial technology for facilitating the integration of renewable energy sources (RES), such as wind and solar energy, into the electrical grid. The challenge of maintaining a balance between incoming and outgoing grid power can be effectively addressed by integrating energy storage technologies with inherently intermittent RES. A range of viable options for storing energy from RES currently exists, among which the Linear Electric Machine Gravity Energy Storage System (LEM-GESS) stands out as a promising choice. The LEM-GESS stores energy in a shaft using piston masses based on the concept of gravity. This paper presents the performance and cost analysis of different linear machines employed as the main drive units in a dry gravity energy storage system. Specifically, linear permanent magnet flux switching machine demonstrates the best performance in terms of overall system cost when considering a 20MW/10MWh system and optimizing for the minimum levelized cost of storage (LCOS). Noteworthy findings reveal that the LEM-GESS cost is highly sensitive to system efficiency, with factors such as material cost and power factor also influencing the LCOS. Designs with modest secondary height, low usage of copper and magnet materials on the primary of the LEM, and a high power factor are preferred to minimize the LCOS. In conclusion, the LEM-GESS with a permanent magnet machine drive option is a highly promising and cost-effective technology for supporting the integration of RES into the grid.