For the sake of narrowing things down, I'm interested in the late Bronze Age, if possible.
I'm familiar with the report by Herodotus that to mourn a cat, they'd shave their eyebrows, or to mourn a dog they'd shave their head/body. But what about to mourn a parent? Or a child? Or the pharaoh? And would there be different degrees of mourning depending on what your place in society was? That is, would a princess mourn in a different manner than a slave?
There are two important things to note about your question. First, your source Herodotus, and secondly a lack of primary sources of everyday life in ancient Egypt.
First, Herodotus's reliability as a unbiased reporter of history is disputed among the history community. Specifically in regards to Egypt, more modern issues have been noted. According to Nielsen, Flemming A. J., "Though we cannot entirely rule out the possibility of Herodotus having been in Egypt, it must be said that his narrative bears little witness to it." Detlev Fehling even goes so far to say that Herodotus never even traveled up the Nile. Given this, it is hard to believe Herodotus's claim that people would save themselves to mourn a dog or cat, especially given the fact that we don't have archaeology to back the claims.
The brings me to the second point. Historians know very little about daily life in ancient Egypt. This is due to the fact that almost everyone in ancient Egypt lived right next to the Nile. Over time any evidence from towns has been since buried under many meters of river deposit. Also, Egyptians had a habit of building new structures on top of old ones. A good example of this is the Middle Kingdom town of Kahun who cannot be excavated due to the fact that it is currently underneath a soccer field.
Given the fact that Herodotus is a poor primary source when it comes to Egypt, and the lack of archaeological evidence of day to day life(which includes mourning practices) means this question probably does not have a sufficient answer.
Sources:
Fehling, Detlev (1994). "The art of Herodotus and the margins of the world". In Z. R. W. M. von Martels. Travel Fact and Travel Fiction: Studies on Fiction, Literary Tradition, Scholarly Discovery, and Observation in Travel Writing. Brill's studies in intellectual history 55. Leiden: Brill. pp. 1–15. ISBN 9789004101128.
Dalley, S. (2013). The Mystery of the Hanging Garden of Babylon: an Elusive World Wonder Traced. Oxford University Press. ISBN 978-0-19-966226-5
Nielsen, Flemming A. J. (1997). The Tragedy in History: Herodotus and the Deuteronomistic History. A&C Black. ISBN 978-1-85075-688-0.