Omnicharge is smart, but sometimes it can be too smart. To save standby power consumption, Omnicharge's Auto Shut-off function will turn off the Omnicharge when the current drawn is lower than a designated threshold. The problem with your Bluetooth headset (or other small gadgets) is that it has a small battery inside and only requires very small current to charge.
You can disable this feature by turning off the 'Auto Shut-off' function from the menu:
Step 1: Double tap the Power Button and enter the menu.
Step 2: Scroll down and enter 'Auto Shut-off'.
Step 3: Choose 'Off' to deactivate 'Auto Shut-off' and exit the menu. All set!
Omnicharge charging a Jaybird F5 Wireless Headphone at 8mA (fully charged).