My mom has a business and mostly uses Windows and Mac OS X; I mentioned Linux because it could help save money.
She is unfamiliar with Linux though; I gave her some basic introduction (uncertain if it's any good, but sure), but I would like to be able to ease her into Linux if she prefers it.
People need to experience something before they can ascertain if it is a viable option. Install Ubuntu or Linux Mint on a secondary device, let her see how familiar it is to her.
Then you can explain some of the other advantages of using Linux.