Yes she can apply for a green card from the United States. If you are a US citizen, she is an immediate relative. Contact an attorney for advice.
I would suggest consulting an attorney to make sure she is eligible before you begin with any applications. That being said, a person can switch from a B visa to another category however there are some issues with what her intent was for entering the country - another good reason to consult with an attorney.
You can apply for her, and don't worry about her visitors visa once she files for adjustment of status.
Mr. Shusterman is a former INS Trial Attorney (1976-82) with over 35 years of immigration experience. His response to your question is general in nature, as not all the facts are known to him. You should retain an attorney experienced in immigration law to review all the facts in your case in order to receive advice specific to your case. Mr. Shusterman's statement above does not create an attorney/client relationship.
Your mother is an immediate relative of a U.S. citizen. She should be able to adjust status in the U.S. The petition and the application for permanent residency can be filed at the same time, You might want to contact an immigration attorney to help you with the filing. There are a few open questions in your case. Did she enter the U.S. with the intent to stay? What questions were asked when she entered the U.S. ? What was said when she applied for the tourist visa?