Your mother will have to qualify for a visitor's visa, and show that she intends to return to her home country following her temporary visit to the U.S. In most cases, if she gets a visitor's visa she will be admitted to the U.S. for a 6-month period.
Right. She would have to apply for a visa in her home country. You could furnish a support letter but it has to be done there. And as the other responses indicate, she would have to show that she has every reason to return after her visit is over.
If you are a U.S. citizen, you can also petition for your mom to become a permanent resident. It may be that if you want child care for more than 3 months, and you want a more stable way for your mom to stay here as long as she wants, pursuing residency for her is not a bad idea. But she has to want to live here, of course.
You can contact me about this at www.lyttlelaw.com