Eloise, FL Immigration Law Firms & Lawyers

With the proper explanation, your mother should be able to apply straight for citizenship without asking for a new green card. She can also explain her situation if she is traveling in and out of the country. If you have some documentary proof (outside of Wikipedia) that the suffix is only a name added as a sign of respect, that might help an immigration officer to understand the situation more quickly. Due to the limitations of the Lawyers.com Forums, Alan Lee, Esq.'s (the "Firm") participation in responding to questions posted herein does not constitute legal advice, nor legal representation of the person or entity posting a question. No Attorney/Client relationship is or shall be construed to be created hereby. The information provided herein by the Firm is general, and requires that the poster obtain specific legal advice from an attorney. The poster shall not rely upon the information provided herein as legal advice nor as the basis for making any decisions of legal consequence.  

You don't indicate your current status, but if you are living in the US and are out of status, your employer can't petition for you. There may be more details you need to discuss with an attorney.

She can legally stay until her visa expires. She can apply for an extension and may receive one for six month, after that she will have to leave the country. If she has a multiple entry visa, she may return at a later date.