With the advancements in underwater wireless communications, internet of underwater things (IoUT) realization is inevitable to enable many practical applications, such as exploring ocean resources, ocean monitoring, underwater navigation, and surveillance. The IoUT network comprises battery-operated sensor nodes, and replacing or charging such batteries is challenging due to the harsh ocean environment. Hence, an energy-efficient IoUT network development becomes vital to improve the network lifetime. Therefore, this paper proposes unmanned aerial vehicle (UAV)-aided energy-efficient wake-up designs to activate the underwater IoT nodes on-demand and reduce their energy consumption. Specifically, the UAV communicates with water surface nodes, i.e., buoys, to send wake-up signals to activate the IoUT sensor nodes from sleep mode. We present three different technologies to enable underwater wake-up: acoustic, optical, and magnetic induction-based solutions. Moreover, we verify the significance of each technology through simulations using the performance metrics of received power and lifetime. Also, the results of the proposed on-demand wake-up approach are compared to conventional duty cycling, showing the superior performance of the proposed schemes. Finally, we present some exciting research challenges and future directions.