I think we should start with what women couldn't do in Egypt. For most of Egyptian history the only thing truly forbidden to the Egyptian woman was being a Pharaoh (they could be Queens) and they were not permitted to be soldiers, particularly infantry or those that managed soldier. Again, the queen could be an exception to that particularly if she was ruling as a regent for a son or nephew. Even despite this being forbidden we do see queens acting as pharaoh and there are historical records of women involved in military endeavors, particularly around overthrowing of invaders to home and country. There are some that contend that women were excluded from bureaucracy. This is highly contested and I will discuss it below. Though not really forbidden you don't often see evidence for women as stone masons or those moving large stones for building. Heavy labor seems to be less common for women.
So, what were the rights of women?
Women could own and inherit property. This meant money, land, businesses and animals among other things. These inheritances were often equally split between brothers and sisters. If widowed, a wife would receive 1-2 thirds of her husband's wealth depending on the era. Women could run businesses as well. Some businesses were strictly run by women for much of Egyptian history. These businesses include sacred mourners and wailers (part of a funeral procession), weavers and those that administered medical care to women and often young children as well. Later men would join these businesses but still most imagery shows a greater proportion of women in these professions.
Women could divorce. Divorce processes were not long and drawn out like modern times. One spouse would claim some misdeed and proof was shown then a decision made. There are accounts of women divorcing husbands because he didn't do his fair share of work to maintain the house or he did not provide enough sex. Men could also divorce in the same manner. Divorce in Egypt was not a legal matter but a personal one, same as marriage.
While there were harems and polygamy in some times and parts of Egyptian history, harem women could keep their possessions, land and businesses unlike other areas of the Mediterranean world where harem women were slaves or an equivalent.
Women played major roles in religion and religious functions. They served as everything from chantresses to doctors, dream readers to caring for sacred animals. Women filled much of the role as sacred dancers and musicians through much of Egyptian history. In the clergy the only gender based limit seems to be that high priests were usually men.
Women could be traders and captain boats. There is quite a lot of evidence for women being the importers of fine goods.
There are some records of women being viziers to the pharaoh. These were usually positions filled by a mother or aunt but there seemed to be no stigma to having a female vizier vs a male one.
It is expressed by some, that Egyptian women were forbidden from bureaucratic jobs. However, there are women with the titles of judge, overseer and director of funerary priests. These suggest that women were in bureaucratic positions in Egypt.
Grave goods showed that there were women capable of paying for their own funerary goods including statues, gold and other materials. This shows not only wealth but also control of that wealth and its use.
Women often ran estates even when their husbands were home. "Mistress of the House" was a common title for Egyptian women. This didn't mean house wife but expressed the fact that the woman managed the finances and goods of the home as well as any production, servants or other persons working there.
There is a lot of support for women congregating for work, social needs and support systems.
While women are mentioned in stelae and tomb carvings with male relatives there are also quite a number made by women that only feature the woman who commissioned it or the woman and her female relatives.
Many women, particularly those who had brothers who were schooled at home or mothers who had been schooled, received the same education as a male child might. This was very uncommon in ancient times.
Women are depicted hunting and trapping birds as well as running marketplaces and workshops.
Women had full access to the court system as witnesses, plaintiff, entering or writing contracts, collection of taxes and estates and so on.
Fathers, particularly pharaohs, often appointed their daughters to important roles in inheritance and religious stations. Often during royal succession the princess had royal blood and it was her presence and will that gave her husband the right to rule.
Women had full access to medical care of an equivalent quality to that of men.
Women enjoyed the same access to food and drink. There are references to images that may be drunk women, including women vomiting from overindulgence in beer.
Single women, especially widows, are referenced as adopting children.
Queen Tiy, wife of Amonhotep III, is often referred to as "common-born" meaning she was probably not of the royalty or nobility but was still sought and married by the pharaoh.
Some titles given to women not mentioned above include: Overseer of the physicians, overseer of the female physicians, director of the dining hall, overseer of the weavers, divine mother, caretaker/overseer of the royal hair (ie the royal hairdresser), Mistress of Amun/Osiris/Isis/Hathor/etc, Chantress, Keeper of the halls (house maid?), Officiate of ceremonies (official of religious celebrations), Mother of Upper and Lower Egypt and so on.
One more brief comment about women. In Egyptian religion and society women had a very distinct role in life. Men were seen as fertile and fertility but the society believed that it was only through the nurturing of women that things grew prosperous. In this belief most agricultural and fertility deities are male but when it comes to nurturing (or the lack of nurturing) goddesses are used. One thing that stems from this is Egyptian ideas about birth. While the Egyptian mind believed the semen was what created life it was the women's body which could nurture that life into a child. With these religious concepts women were seen as nurturers of society. Anything that needed to be cared for money, goods, children, the dead, etc were often considered in the domain of women.
-Kelsey Museum Exhibit on Gender and Ancient Egypt
-The works of Barbara S. Lesko including the research paper "Women's Monumental Mark on Ancient Egypt" from The Biblical Archaeologist and books: The Great Goddesses of Egypt and Women's Earliest Records: From Ancient Egypt and Western Asia
-"Women of Ancient Egypt and the Sky Goddess Nut" by Susan Tower Hollis
-"Everyday Life in Ancient Egypt" - By Lionel Casson
- Kahun and Ebers Medical Papyri