Can I get a green card for my mother's boyfriend?

I believe that it would prove very difficult for him to obtain a visa based on the situation that you have outlined.  The fact that your mother's boyfriend is sick is detrimental in a visa application adjudication. The Trump Administration has recently even attempted to deny extensions of deferred action for people that the US previously admitted for medical reasons. Other negatives are that he never married your mother (if the request is for a humanitarian parole) and his longtime stay in the US for 11 years if he was illegal during that time. 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.