From the video it sounds like the noise could be the result of something underneath the power button keycap, but that would not explain your other symptoms. Where does the noise seem to come from within the machine? Is it loudest near the power button itself or near some other component? The only other moving part in your system is the fan, so it might be a good idea to remove the bottom panel, open the machine on its side, and observe the fan when hitting the power button.
Another possible theory is that when you're trying to power on the machine, the system is actually already on but in a frozen sleep state. So in other words, when you hold down the power button to start it you're actually forcing the frozen system to power off (Mac has a feature where if you hold the power button for about 5 second it forces the machine to power off). Forcing the machine to power off like this sometimes produces a click noise (I think this is because the fan suddenly stops). You can check to see if the system is actually on by pressing the CAPS lock key and seeing if there's a green LED on the keycap.
I know some of these models have sleep/wake issues after installing third party drives, I think it has something to do with the firmware. If you think that might be the issue let me know and I'll do some research to see if there's a way to resolve that.