I am going to assume that your friend is a US Citizen. If not, then he cannot file an I-485 for her. If he is USC, then yes, he may file an I-485, I-130 and all other relevant documents. Once they receive the NOA. His mom will be in a status called "AOS pending" and it is legal until she receives "approval/denial" from USCIS.
Make sure you to file, at least 1 week, before her I-94 expires. Not that it will cause any problem as she is an immediate relative of USC. But to avoid additional "$1000 fees" and more questioning during the interview. It is better for him to apply early.