My daughter recently purchased a house and the mortor in the chimney has mostly crumbled away and several bricks are missing. Should I just try to replace what is bad, or totally redo the chimney?

Depends on the age of the house and the kind of brick used, but probably, all that needs to be done is to replace the bricks, and then re-point ALL the mortar joints in the entire chimney.

If the chimney was made of common brick, and the brick themselves are crumbling, it's a different matter. Common brick were never meant to deal with weather; they were to be used only inside of a thick brick wall. The brick that are meant to be on the exterior are called face brick. The difference between the two is that common brick are porous, and face brick much less so.