I found old ( what I think is BX ) flexible metal cable containing cloth covered wire ( two wires ) behind the medicine cabinet I am replacing. There was a splice in the middle. But the splice was only taped. I would like to replace this old wiring. Some places I can get to. Others I cannot. Is there a way to pull new wire?

Old house plaster and wood lathe.

Any suggestions?

If you have attic or underneath access, It will make it much easier. Many times you can attach new wiring to old, pulling in place of. Very old houses usually sometimes no or little
Use of staples. You may have to remove switch boxes, light boxes, and receptacle boxes to fish new wiring in wall, recessed cans make convenient way to get wires around.