I don't know if there are any Egyptian sources talking about it, but Herodotus wrote about it in 440 BC:

The animals that do exist in the country, whether domesticated or otherwise, are all regarded as sacred... The inhabitants of the various cities, when they have made a vow to any god, pay it to his animals in the way which I will now explain. At the time of making the vow they shave the head of the child, cutting off all the hair, or else half, or sometimes a third part, which they then weigh in a balance against a sum of silver; and whatever sum the hair weighs is presented to the guardian of the animals, who thereupon cuts up some fish, and gives it to them for food- such being the stuff whereon they are fed...

If a cat dies in a private house by a natural death, all the inmates of the house shave their eyebrows; on the death of a dog they shave the head and the whole of the body.

[The History of Herodotus. Trans. Rawlinson. Book II. http://classics.mit.edu/Herodotus/history.2.ii.html]

Shaving your eyebrows was a sign of mourning, like wearing plain white or black clothing in Medieval Europe for X period of time. The eyebrows served as a visual reminder that somebody was in mourning, and once the eyebrows grew back, that person was no longer in mourning.

Just remember that Ancient Egypt was a diverse culture spanning thousands of years, so you can't necessarily assume that this practice happened at any other time or place than the one which Herodotus visited. At least I can't with my limited knowledge.

And note that Herodotus wrote based on second hand knowledge, so you can't treat it as irrefutable truth. His histories contain some facts and some myths.