Omega2 requires solid 3.3V stable voltage output at 300mA.
And one more critical spec... power supply needs to be able to handle in-rush current during WiFi power-up without voltage drop.
That fleabay $0.99 item needs a large capacitor (I recommend 220uF 6.3V) on its input side and a large capacitor on its output side (I recommend 470uF 6.3V.) The link between that unit and Omega 2 cannot be too thin gauge (I recommend 22 ga) and definitely use soldering as opposed to using DuPoint connectors.
Added some more pictures as requested in ticket 16582.
When omega failed, it was connected like this – nothing more was connected.
Later on, as recommended in ticket 16419, serial connection was added to the header pins 5 & 6 (GPIO 13 & 12) and I'm sure I wired it correctly because I checked on other omega I have and I saw omega ASCII art and the kernel loading: