And how did that name come about ?
It comes from the Navy when the useless fray end of rope was called the fag end. In cigarettes the filter is referred to as the butt or fag for the same reason it's useless, but the whole cigarette ended up being called the name of the filter. This is why people ask if can they "Bum a fag?" which is to share a cigarette and get most of the filter end, or to stick the filter in the mouth and make it wet. Other names for cigarettes are "smokes and stokies."