It means take another pain pill before her next dose is due, so she 'doubles' the pain medication she is taking. 'Double dip' comes from dipping chips into a dip. It means dipping the chip, biting off the piece with the dip on it, and then dipping the same chip again. It's considered unhygienic and rude. You should eat the whole chip and dip a new one. But here it just means take an extra pain pill in between the two she has been told to take by the doctor.

The traditional meaning of ‘double dip’ is to take more than you are allowed to, and had a generally negative connotation. For example, charging someone for expenses incurred even though they had already been reimbursed, or collecting insurance from two policies even though there was only one loss. Now it means two of something when you only need one, and can convey an idea of decadence. For example, double-dip fried chicken. In my opinion, your example isn’t a very good use of the term. We say things like ‘I doubled up on the pain meds.’ This is especially true because ‘double dip’ now has the ‘chip-dipping’ meaning as well.