Flashing Power: If I press the power button on my Ouya. It only flashes for a moment. If I press and hold the button it will begin to power up until I remove my finger.
This problem I've been hearing it pop up from a handful of users but never really experienced it myself which is the flashing power issue. In this scenario, it is assumed that you get at least some form of video on your screen when you are holding down the power button such as the orange (U) logo upon power-up. Perhaps if you hold it down long enough it comes all the way up to the main menu. If you are not even getting video then it may be something more serious then a simple power switch issue.
The push-button on the Ouya is not a hard power on/off switch like what one would expect. The button simply tells the processor if it is in sleep mode to turn on. Now usually when you plug the power barrel plug into your Ouya in it automatically turns itself on without even hitting the power button. Without having one of these defective units in my hands I can only theorize that perhaps there is a MOSFET that the push-button controls which is not remaining in its closed state after it is fully powered up. Or, the Tegra 3 processor is misfiring and polling the switch status as off when it should remain on until the switch is otherwise pressed again.
If you have this problem and you bought your Ouya within the year contact Ouya to open a helpdesk ticket to get your unit replaced.
If it's out of warranty and you have some soldering skill. I'd just install a single-throw toggle switch into the case and then solder that to the contacts on the momentary push switch. If the Ouya wants to sense the circuit is closed or else it won't power. Then give it exactly what it wants. After you go through the Ouya update or reflash the firmware and it may go away completely afterward!
12/25/2015 - This happened to one of my Ouyas finally and all I had to do was replace the power supply on it. The power supply had an unstable 12v power going into the Ouya causing it to freak out when it began to power up.