Without seeing the exact circuit, it is hard to say.
A capacitor could be used for numerous things on the board. Just because it is near the outlet cable does not indicate it is being used for circuit "power". When you state it is "next" to the outlet cable I assume it is not directly connected to the "outlet cable"; therefore, it could be used for anything.
A capacitor is most commonly used as a filter. It can "smooth" a rectified voltage ripple to create a nice looking DC voltage, or it can be used for noise reduction.
If it is a smoothing capacitor, you could put the same or larger size capacitor, with the same or larger voltage rating, in place of the toasted one. Chances are, you are using a Wall-wart ( the ugly black box that connects to the AC outlet), and the smoothing would be taken car of there, so I doubt it is a smoothing capacitor.
In the case of a filter, it could clean the input voltage and would be a simple capacitor connected from Vin to GND, to reduce external Noise. Or it could be a low/high/notch filter. In either of these methods, you should replace it with the same value of capacitance(parallel combination works too).
Chances are pretty good that the capacitor is not the only source of malfunction. Check for a fuse or a burnt trace.