For the green card application, do we use her name on the certificate or her married name?

In the process of applying to become a naturalized citizen, your wife may (but is not required to) have her Naturalization Certificate identify her by her married name. If she changed her name in a way other than simply through marriage, such as through a Court Order legally changing her name, then she will be required to inform the USCIS of that and to supply a copy of the court order; her new legal name will then appear on her Naturalization Certificate. Inconsistent identification of a person's name in important documents can add significant complexity to an application process. Nonetheless, regardless of whether your wife's name change makes her birth record, naturalization certificate and/or other relevant documents inconsistent, she still may be able to sponsor you to adjust status to become a Lawful Permanent Resident (to get a "Green Card"). It would be wise for you and your wife to consult with an immigration attorney who, after learning all of the relevant information, could advise about eligibilities, options and strategies to attain your goals, and who then could offer legal representation in the often complex application process.