The Schengen system merely aims at removing systematic checks at the border. As discussed in Sufficient Identification for Intra Schengen flights it does not particularly address ID requirements for air travel. It's not even clear that it really entitles you to travel without ID as some countries make it mandatory to carry one in any case and do perform occasional checks.
As I reported in my answer to the earlier question, you might get lucky and be able to fly with other documents but presenting a national ID card or passport still seems to be the norm, certainly at Schiphol. An expired passport, if she has one, would also generally be enough.
Unless you have to fly right now, the best would therefore be to get in contact with the relevant consulate and try to obtain some sort of emergency travel document.
The EU website also suggests as much:
- lost your passport or had it stolen?
- realised that your passport has expired during your trip?
In either situation, under EU rules you may travel only with a valid ID card or passport. But help is at hand, as the EU countries have systems in place to deal with such cases.
The conditions and procedures do vary widely from country to country. So if you're in the EU, your first port of call should be your country's consulate or embassy.
Alternatively, she might consider going by train or some other means of overland travel. It does not make it legal or fully guarantees that there will be no ID check but the risks would seem lower and even if there is a control, she could hope to resolve the situation without entirely losing the benefits of her ticket.