This work proposes a framework to design a cost-efficient unmanned aerial vehicle (UAV)-based energy-neutral (EN) system deployed to harvest data from a set of internet-of-things (IoT) nodes. The energy-neutrality refers to the zero-sum balance between energy harvested, stored, and consumed during operation, which is a game-changer when a connection to the electricity grid is not available/feasible. This involves employing an off-grid charging station (CS) comprising of photovoltaic (PV) panels and batteries that provide enough energy to recharge the UAV-based aerial access points (AAPs). The investment cost is determined by the number of AAPs, PV panels, and ground battery units. Its minimization cannot be achieved using conventional optimization tools due to the non-tractable form of the CS load. Therefore, a novel wave-based method is proposed to represent the load profile as a proportional function of the required number of AAPs, so as to directly relate the CS design to the trajectory optimization. Compared to baseline scenarios, the proposed trajectory design can halve the time and energy consumption; the investment cost varies with the time and season of service; the off-grid CS is particularly advantageous in rural areas, while in urban areas its cost is comparable to that of a grid-connected alternative.