She could try for a student visa.
Unfortunately, the turmoil in Greece isn't a 'problem' that the US Government feels responsible for. Thus, you/she will find it best not to mention this when applying for visas, or consulting with people.
Consider a paid consultation (many of us use Skype for this) with an immigration attorney ... to explore other options.
PROFESSOR OF IMMIGRATION LAW for over 10 years -- This blog posting is offered for informational purposes only. It does not constitute an attorney-client relationship. Also, keep in mind that this is an INTERNET BLOG. You should not rely on anything you read here to make decisions which impact on your life. Meet with an attorney, via Skype, or in person, to obtain competent personal and professional guidance.
The situation is Greece isn't bad enough to give your cousin any special treatment for staying in the US beyond her authorized time. If she entered the US on the visa waiver program, she really has no choice, but to leave the US. She can't change her status to student in the US, but she could apply for a student visa back in Greece.
This general advice does not create an attorney-client relationship.
Options depend on facts. You really need to meet one on one with an experienced immigration attorney, whether myself or one of my colleagues, so that he/she can review all your documents and facts in order to determine what your options are.
J Charles Ferrari Eng & Nishimura 213.622.2255 The statement above is general in nature and does not constitute legal advice, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship. Answers on Avvo can only be general ones, as specific answers would require knowledge of all the facts. As such, they may or may not apply to the question.