How come my mom was denied a re entry visa?

A reentry permit must be applied for while the applicant is in the United States. Since your mother is overseas, she has choices of trying to apply for a special immigrant visa with the American Embassy or boarding a plane back to the US and explaining with documents her situation to Customs and Border Protection at the port of entry. Immigration lawyers by and large suggest the second course of action since it appears to be more difficult to obtain a special immigrant visa. Due to the limitations of the Lawyers.com Forums, Alan Lee & Arthur Lee, Esqs.’ (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.