N., the most common cause of dark circles under the eyes is not fatigue, but allergies.
If your daughter is having a stuffy nose, itchy eyes, or difficulty breathing, seeing a pediatrician and then a pediatric allergist is a good idea. It's imperative if you see difficulty breathing, even a slight amount. The pediatrician can diagnose allergies generally and prescribe medications, but the pediatric allergist can identify the offending allergens and set up a more complete course of treatment. I would not treat with OTC meds until you are sure of allergies, because it is not good to use antihistamines and decongestants for no reason in children. If she is getting allergies, they will bother her and distract her in school, so it's important to get this sorted out soon.
If she is getting enough sleep, then I would look to allergies for the cause.
M. ( had allergies all my life, even from infancy, know what it is to have them in school)