How do I get my 84-year-old mom to give up smoking?

Getting someone that old to stop smoking is like trying to get someone of any age who has schizophrenia to stop smoking. Even if you tell her your family's asthma problems, she probably will not quit. Sounds like you are going to have to tell her that for the sake of your family's health other arrangements will have to be made other than coming to live with you.

Sorry to say that you probably can't get her to give up smoking at 84. I also suffer from allergies and cigarette smoke is especially bad for me. If it were me, I could not have her in my home; I'd have her move to assisted living. Generally, these places have some smoking rooms or smoking areas but allow no smoking in rooms or apartments. This will ensure she is safe, gets care / assistance she needs and you're all healthy enough to visit regularily. Best of luck.