A.L.
To me, this sounds for all the world like a sinus migraine.
Is your friend sure her daughter is truly tired? Or is she sleeping a lot because, like a lot of us, this is the only time she has relief from the headache?
For pressure to cause a black eye, it doesn't always have to come from the outside. It can come from swollen sinus cavities, too - and mine, like this little girl's, typically swell only on one side, and is generally caused by allergies. Personally, I would not call this "mild allergies" - I would call it severe seasonal allergies. But still allergies.
Staying hydrated will help - she won't necessarily have a runny nose - I don't always - but staying hydrated will help keep the congestion from getting quite so thick. I would, however, expect her to have an occasional sore throat, as the sinuses drain.
I hope she finds some relief, no matter what the cause.