Is it permissible for a sister to go out to work, despite not being in financial need of it, given the current security conditions that raise fears of abduction, and her strong insistence might be due to feeling confined from staying at home, or is it better for her to stay at home and only go out when necessary or to a Quran memorization center?

You have done well in preventing your sister from working outside the home, as there is no necessity for it, and there are many temptations outside. You must insist on your position, while being kind to her and advising her with goodness, and clarifying that your prevention is for her benefit. Emphasize to her the importance of joining Quran memorization classes and associating with righteous women. Mention to her the sayings of scholars that support this, such as the statement of Ibn Baz that a woman's staying in her home is the norm and best for her, and the statement of Ibn Uthaymeen that a woman's staying in her home is not imprisonment but rather joy, modesty, and avoidance of temptation. Treat her depression by taking her out for recreation and family visits. If these means are not effective, look for another job for her close to home and free from mixing with men, while ensuring her safety from potential dangers.