I'd guess you are in your 20's, hence the wise mouth.
And I'd say she choose the wrong person to ask, based on what you've written here.
She needs a few classes, and some training, and of course range time. Then practice in her home. She doesn't need much, because she is interested in her own home, not unfamiliar buildings. She knows her home, an intruder most likely will not.
Suggest she get interested in some basic training. Pay her way into a beginners firearms safety class, encourage her.
The Mom you save may be your own.
BTW, she is not "cheap". She is frugal. She tries to spend her money as wisely as she can.
Other folks I have known who were actually well off were as well, very frugal. Nothing wrong with that, they probably worked very hard to get where they are.
I bet your Mom worked her butt off, too. And what she gets is disrespect for it.
A well placed shot with a .22 will most certainly incapacitate or kill an intruder. But untrained people
get killed with their own weapons every day.
Take a class with her, encourage her.
Don't be disrespectful of her. She's done the best she could.