Hi bobono
Your granddaughter could be experiencing elevations in her temperature due to her menstrual cycle, if she has one already, and if not, it could be that she is premenstrual....meaning that she is getting close to having her first. If there are no other symptoms that she is experiencing, and they are not HIGH elevations, I would be hard pressed to say to treat them with an antipyretic (fever reducer). There are many girls who start their periods at her age, unfortunately for them. Is her development ahead of the expectations in puberty? If so, it's a good indicator as well that she could be peri-menstrual! Good luck!
I would not give her a fever reducer unless her temperature was very high, and then only under supervision of a physician. Remember the adage "mopther nature never does nothin for nuthin". Fever is part and parcel of the immune system - most times - but every once in a while the organ regulating temperature in the brain sends out mixed signals, sometimes due to a tumor. You need to keep searching fotr a physician who will pay attention and get to the bottom of the reason for the re-currant fevers. I urgue you to obtain the two books "How Doctor's Think" and "A Second Opinion", both by Dr. Jerome Groopman. Reading these books will give you confidence in dealing with the medical establishment.
Keep a journal. How much is the temperature elevation? When does it occur? (Specific date and time of day.) What was she doing when it occurred? What had she been doing the day before? What has she eaten? How does she feel at the time of the temp elevation? These are some of the kind of things the doctor would find of help in making his determination.
It may be that the temp elevation is of such short duration and/or minor elevation that it is of little concern in a nine-year-old.