My cousin is under the Prosecutorial Discretion, can she authorize to visit her home country?

It may be possible for her to apply for an "advance parole" document in order to travel and return. DO not attempt to do this without the assistance of an immigrant attorney-- the consequences of error can be dire.

Your cousin may be able to request advance parole depending upon the basis for her prosecutorial discretion. However, if your cousin was in removal proceedings which have been administratively closed, she is likely ineligible for advance parole. Your cousin should consult an experienced immigration attorney before taking any steps to seek advance parole and/or travel outside the United States.

She may be eligible for advance parole, depending on the reason for her visit. She should consult with an immigration attorney in order to avoid any potential problems. Best of luck.