My US friend wants to spend 90 days in Australia though the remainder of her ETA expires after 56 days?
It looks like it may be possible. She'll need a new visa, if she doesn't want to leave and come back though.
According to the Find a visa page, she could apply, in Australia, for either a Visitor Visa (subclass 600) or a Work and Holiday Visa (subclass 462). They both indicate that she should be able to apply: from within Australia, online, for up to a year.
I would strongly suggest she read all of the notes on that page and figure out which would be best for her.