Is it more befitting for the engineer to work for two years to secure the costs of Hajj for herself and her husband, or to stay at home to raise her daughter with an Islamic upbringing?

The default is for a woman to remain in her home because it is better for her and more protective of her honor. Allah Almighty said: "And stay in your houses, and do not display yourselves as [was] the display of the former times of ignorance." A person is not obliged to seek to acquire the expenses for Hajj. There is no harm for a woman in working, provided it aligns with the Sharia's guidelines and does not lead to the neglect of an obligatory right. If a woman does not have a need to go out to work, then it is preferable for her to remain in her home to manage the affairs of her home, her husband, and her children. Raising children, in particular, is invaluable. However, if she finds work that suits her nature, there is no harm in her practicing it. She must also be keen on raising her children well by placing them in an environment that upholds sound upbringing upon the creed and ethics of Islam, and not neglect them.