Does her overstayed visa status disqualify, or hurt her chances of obtaining a green card?

Generally, a foreign national who entered the U.S. lawfully and with inspection, such as with a student visa, and who then becomes married to a U.S. citizen, may apply to adjust status to become a Lawful Permanent Resident (to get a "Green Card") regardless of whether she may have violated or overstayed her student visa. No petition for a waiver is necessary in that process (but note that until the couple properly initiates the adjustment of status process, if the former student were to come to the attention of immigration authorities she may be subject to being placed in removal/deportation proceedings). The adjustment of status application process may be more complex than it appears at first blush, and it would be wise to work with an immigration attorney.