What will help clear the air in my Mom's house?

My Mom is a HEAVY smoker, has been for years. When I step into her house, literally, my eyes hurt, and I can barely breathe. I ask her why she doesn’t open the windows, but she reminds me that it is winter, and she can’t afford to have the windows open. I feel bad that she is constantly breathing in not only the direct cigarette smoke, but the second-hand smoke that fills her house daily as well.
I was thinking about getting her some kind of air purifier, but I’m not sure which kind. The terms they use confuse me…ionic, negative ions, etc. So maybe somebody has used one, or knows something about those types of products. If you do, please help me. I just hope it’s not too expensive, whatever you all recommend, as I am on a budget. I think this will make a good Christmas gift. Haha, thank you