My cousin came to usa for a 3 month visit from Greece. For medical reasons, she must stay another 3 months before returning to Greece. She came on an ESTA visa. How can she get a 3 month extension to stay in the USA? She is living with me and speaks very little English.
It sounds like what you are actually talking about is extending the 90 day period that you are allowed remain in the US when entering under the Visa Waiver Program (VWP), which the ESTA is part of the process for.
In theory, what you are asking for is not possible.
However, it IS possible to obtain a single 30 day extension in the event of medical or similar situations, however it is at the discretion of the USCIS (US Citizenship and Immigration Service) as to whether this will be approved. The process of requesting an extension is documented on the Customs and Border Patrol Website
If longer than 30 days is necessary then the process is going to be - in the first instance at least - the same as above.