Someone from Denmark is here on the visa waiver program (VWP) to take care of her terminally ill sister.
What happens if she reaches the end of her 90 day period of admission and she still needs to remain here to take care of her sister?
She has to follow the law.
So either she has to LEAVE the country, and return again on another 90 day set. This may raise concern from immigration officials, but if she can prove she's not working in the US, and has a job/life to return to in Denmark, it is permissible.
Note, however, that she cannot reset the 90 days by going to Canada/Mexico. You need to leave the continent.
Alternatively, she'll have to apply for a different, longer visa. At this stage it'd be worth talking to an immigration specialist, as it can get quite tricky.
Final option - if it's an option at all, for the sister to go with her to Denmark. Of course this depends on her ability to be moved.
I am not a lawyer, or a VWP expert but I found this information on myattorneyusa.com
It is possible to extend your stay past the allowed 90 days in the Visa Waiver Program but only in case of an emergency. Keep in mind that any extension is not to exceed 30 days. This extension, if approved, will not have any negative impact on the Visa Waiver requirements and thus, you will not be considered to have violated the terms of the Visa Waiver Program.
As for the emergency on which this request for an extension is based, it needs to be legitimate and “real”. The Customs and Border Protection website mentions emergencies that are allowed, such as hospitalization, weather conditions which prevent air travel or in some cases, worker strikes. Absent any of these or similar emergency situations, United States Citizenship and Immigration Services (USCIS) may likely deny the extension request.
To request this extension of stay, you should schedule an InfoPass appointment with a USCIS officer using USCIS' website, uscis.gov. At this appointment, you should provide documentation and any other evidence to the USCIS officer to support your need to extend your Visa Waiver stay. It is strongly advised to keep any and all records that support your request. These records may include canceled airplane tickets, statements from doctors or airport personnel or any evidence showing harsh weather conditions.
I don't know if the described circumstances would count as an emergency or if a 30 day extension would be sufficient but making an InfoPass appointment and discussing the situation with a USCIS officer would seem a sensible way to attempt it. Even if the officer says the sister must leave the USA, they may be able to advise on how quickly a return VWP visit would be allowed in the circumstances.
To complement the great existing answers, I'll briefly point out what happens if you overstay the visa waiver program (VWP). The information is taken and quoted from here:
1) Bad news: You will not be eligible for future entries under the VWP, but have to apply for a B-2 tourist visa (and pay the applicable fees) at the U.S. embassy or consulate. At this point, you will need to provide plenty of evidence to prove your "nonimmigrant intent" (that you intend to return to your home country) and that you can afford your trip to the U.S., such as an itinerary showing your return date, proof of your permanent job or residence in your home country, and financial documentation.
2) Good news: You will not be subject to a time bar on re-entering the U.S., if your overstay did not last more than 180 days. Make sure that you have proof of your date of departure (passport stamp or plane ticket, for example) in order to prove that your brief overstay shouldn’t trigger a time bar.
I'll gladly correct these points if anyone has additional insights/corrections!