This is a great question, but a little difficult to fully answer.
If we are concerning ourselves only with strict grammatical correctness, then yes, your mom is correct. However, do people say "I am staying home."? Yes! They certainly do. Who says this? EVERYONE! So, is it incorrect if everyone is saying it?
Every language has two forces working on it, a force to preserve it and keep it "original", and a force to change it, to evolve it, to bend it, to stretch it, to reshape it. Your question is an example of a moment in time when a language is caught between those two forces. But as much as some of us wish to preserve language (usually the older generations like your mom and me), change always wins in the end. Languages always change. They are liquid and reflect the fluidity of our changing lives.
Your example is very similar to a couple that bother me a little - the disappearance of "that" and "of" in certain language constructions. For example, people say this quite naturally "This is something I don't like/like/love." But what happened to "that"? We "should" be saying, "This something THAT I don't like/like/love." There are many, many examples where "that" is being left out of language. I suppose it's because the idea of "that" is understood.
Another example that really bothers me is the absence of "of". Everybody says "all the time". For example, "He loves the beach. He goes there all the time." But it should be "all of the time". One of the functions of the word "of" is to show portion or amount - a cup of sugar, a slice of pie. Expressing an amount of time is no different. We would never say "some the time", or "none the time". You would sound strange if you did. We say "none of the time", "some of the time". "I'm not as thrilled with the beach as he is, but I do go some of the time".
So, which is right: staying home or stay at home, all the time or all of the time? It's difficult to say, but it's a great way to have an interesting discussion with mom.