Does an 18 year old who had her baby adopted when she delivered. This was arranged by her doctor. No one in the family knew she was pregnant and now a week later she is having such remorse. Does she have any rights? Is there a statue of limitations?

Depends on where you live. In many states, no, there is no statute of limitations. Some have a 4 days window, some more.

Did she sign away her rights in the delivery room? Has she even been to court to relinquish her rights yet?

There are a LOT of laws that protect birthparents.

If she hasn't gone to court to relinquish her rights, then she needs to get a lawyer TODAY and pursue this.

If she DID relinquish her rights in court, then your local laws on changing her mind were explained there--but again, she'll need a lawyer.

I don't know of a single state that allows relinquishment outside of a courtroom---and ESPECIALLY not in the delivery room.

Get a lawyer.