What is this electronic component, in series (I think) with the fuse just on the input of a computer PSU?
It's likely an inrush limiting NTC thermistor.

Here's a picture of one from Amphenol:



When you turn the device on the thermistor is presumably cold, and has a higher resistance.  It therefore limits the current (most likely that current is charging big capacitors) and reduces the chance of blowing breakers or fuses.

Once the device has heated up, the resistance drops to the point where it doesn't affect circuit operation, or the losses incurred are acceptable.

Of course if you cycle power on and off quickly when the device is hot, there could be a problem.

Not sure what "low power electricity" is in your context.  If there was just enough current to keep the device in a high dissipation state it could conceivably blow, but without more detail it's hard to say.

It's also possible that there's something downstream that has soft failed, and the thermistor blows due to the increased load.  (Which isn't quite enough to blow the fuse.)