Answer: shaving the eyebrows to mourn the passing of their cat, is a clear signal to others. Some background information on the relationship between ancient Egyptians and animals helps, I think, to understand the practice of shaving eyebrows on the death of a cat at that time. The quote comes from Herodotus, an ancient Greek historian of renown:
“Egypt, though it borders upon Libya, does not very much abound in wild animals, but such as they have are one and all accounted by them sacred, some of them living with men and others not….About these animals there is a custom of this kind: — persons have been appointed of the Egyptians, both men and women, to provide the food for each kind of beast separately, and their office goes down from father to son; and those who dwell in the various cities perform vows to them thus, that is, when they make a vow to the god to whom the animal belongs, they shave the head of their children either the whole or the half or the third part of it, and then set the hair in the balance against silver, and whatever it weighs, this the man gives to the person who provides for the animals, and she cuts up fish of equal value and gives it for food to the animals.”
What he is basically saying is that at that time, in ancient Egypt, there were certain people appointed to make vows to the gods of animals and this role was handed down the family line. When they made these vows they shaved the head of their children. They weighed the shaved hair against the weight of silver. The value of the silver was given to the person who owned the animals. That is my interpretation but you may have your own. And now with that in mind….(source: classicwisdom.com).
Clearly there was a concept of shaving hair from human bodies and this is extended to shaving the eyebrows as a sign of mourning on the passing of a cat companion. It was a signal to others that the person was in mourning. When the eyebrows grew back he or she was no longer in morning. I recall Queen Victoria wearing black for the rest of her life after the passing of her dear husband Prince Albert. This was a very similar form of human behaviour.
Herodotus was a historian and his writing was based, as I understand it, on second hand knowledge and therefore there may be some divergences from fact and the stories may contain some myths. And his writing relates to ancient Egypt at a certain time. He wrote the Histories in 430 BC. It is a record of the ancient traditions, politics, geography and clashes of various cultures in Greece, Western Asia and northern Africa at that time. He is considered one of the greatest storytellers. His record is not completely impartial but it is considered today as a most important source of information about that time (source: Wikipedia).