Possible to identify burnt component and its value?
My supposition:

It is a ferrite bead, and it is being used, along with the capacitor next to it, as a small filter on the power supply.

It is a 'non critical' component and only really has any affect when you're using a noisy power supply.

You should be able to replace it with pretty much any 0603 ferrite bead. Lower DC resistance is better as it reduces the voltage drop across it.

You could even replace it with a blob of solder to get it working again.