If its a total black screen and it happened completely random i.e. not after installing any wads or anything then its probably either the bluetooth module or the wifi module. The bluetooth is the more common one to go and is roughly £8 online. Fitting it will require disassembling the wii most of the way as the bluetooth module can't be accessed by removing just the casing and drive
what i discovered happens to my wii is that it overheats when off due to the Wiiconnect24 Standby Connection. While overheated it doesnt respond when i try to turn it on. while im sure this is isnt your issue, i thought i would just add my insight
I can say from experience this has fixed almost ALL my wii's problems XD my wii used to over heat so badly so now every week a leave it unplugged for a day in a ventilated room and BAM it works again. Hell my wii's DVD drive would scream bloody murder when it was turned on now that i plugged it out for a month (various reasons) it sounds perfectly.