Internet of Things (IoT) devices are largely resource-constrained embedded devices with limited user interface and battery capacity. As a consequence, bootstrapping a secure connection between an IoT device and a wireless network (e.g., WiFi network) becomes a challenging problem since the traditional Pre-Shared Key (PSK) based authentication cannot be directly applied. Proximity-based device authentication is a promising mechanism to enable secure pairing of an IoT device to a wireless network. However, existing solutions do not deliberately consider energy-efficiency nor the tradeoff between energy consumption and security strength in the pairing process. This paper fills this gap by enhancing the energy-efficiency and studying the tradeoff between energy consumption and security strength of an existing proximity-based IoT device authentication protocol, called Move2Auth. An optimization problem is formulated to minimize the energy consumption incurred by the pairing protocol while satisfying the desired security performance. Experimental results based on Raspberry Pi devices show the energy-efficiency advantage of the proposed scheme over the existing one.