The capability of free-space optical (FSO) communications in delivering very high data rates and the agility of unmanned aerial vehicle (UAV) flying platforms render FSO-UAV-based solutions attractive for delivering 5G wireless communication services. In parallel, research on simultaneous information and power transfer, whether in the context of radio frequency (RF) networks or indoor optical wireless communication (OWC) networks, is on the rise. Even though the operation of a UAV is limited by its battery lifetime, the concept of energy harvesting (EH) from the information-carrying FSO signals was not deeply investigated in the literature. This paper highlights the inherent EH capabilities in FSO transmissions and investigates novel signal design methodologies for boosting the EH efficiency. We focus on the ground-to-air communications where the ground-based FSO transmitter is connected to the power grid and, hence, is governed by a peak-power constraint and not an average-power constraint. For this setup, simulations carried out under different weather conditions demonstrate that high data rates can be associated with significant amounts of harvested energy using simple transceiver architectures.