India does not allow dual citizenship hence the requirement of the PIO card you mentioned. But please note that the PIO card is being replaced by the OCI card; and, by October 2018 you will not be able to travel using the PIO card anymore.
Last date of application for free conversion of PIO Cards to OCI Cards
is on or before 31st December 2017 Thereafter a fee of US $275.00 will
be charged as for issuance of a fresh OCI Card; and all PIO Card
holders are, therefore, requested to apply for converting their PIO
Cards into OCI Cards, free of any Consular Fee, on or before 31st
December 2017; The last date will not be extended beyond 31st December
CKGS Service charges & optional value-added service charges (if opted)
With effect from October 2018, International Civil Aviation
Organization will accept only machine-readable travel documents and
hence the existing PIO cards will become invalid. All PIO Card
holders are, therefore, requested to apply for converting their
existing handwritten PIO Cards into machine-readable OCI Cards.
However, overstaying in India is illegal (punishable by fine, imprisonment or deportation) whether it's for 1 day or 100 days and the first thing you should do is to notify the nearest US embassy/consulate and follow their instructions for your passport renewal. The first thing that I suggest is to get her passport renewed. It will be the easy part.
The harder part, then you should follow the path of the OCI card but please keep in mind that you will have to pay fines because of the illegal overstay and late visa renewal. AND she must have a valid visa to obtain an OCI card. It would be a time and effort consuming process because of all of the "official procedures" of the FRRO/BOI for the visa renewal under such circumstances.
On the other hand, getting an Indian passport, hence renouncing US citizenship would be a totally different story.