Women have a special connection to Chanukah. The Shulhan Aruch (siman 670) brings down a custom that women do not engage in melacha activity while the Chanukah candles are burning. This refers to the first half hour after lighting, which is the primary misva. This minhag is also quoted by the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) in parashat Vayeishev, Hacham Ovadia and Hacham Ben Sion Abba Shaul (Israel, 1923-1998).
The primary melacha that should be refrained from is laundry. This includes drying the laundry, weaving, sewing and needlepoint. It is permissible for women to bake, cook and fry. Washing dishes is also permissible. Furthermore, Hacham Ovadia writes in Chazon Ovadia that it is permissible for them to write during the half hour period.
Hacham Ben Sion was asked whether women are allowed to activate a washing machine during the first half hour. He tends to be lenient, but then he writes that since the custom applies for only one half hour, it is better to refrain.
According to the Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933), the reason for the custom is to prevent women from using the light of the candles, which is prohibited. Refraining from melacha serves as a reminder to abstain from their light. The Ben Ish Hai presents a different reason according to the Kabballah.
Rav Haim Palachi (Turkey, 1788-1869) cites a different custom, according to which, women were careful not to work the entire first and eighth day of Chanukah.
Hacham Ben Senyar, in his sefer Ner Sion (Ch.3, Halacha 29), writes that women should wear special clothes on Chanukah, like they wear on Rosh Hodesh and Purim. He adds that if they have gold jewelry, they should wear it during these days. Obviously, women must still be careful about the laws of modesty when dressing up. He cites the Pele Yoetz who promises that a woman who is careful about modesty merits to have upright and holy children, and will reach a ripe old age in health and happiness.
Finally, the Ben Ish Hai brings down a minhag told to him by Rav Eliyahu Mani: It was a custom of women, if, G-d forbid, they found themselves in a dangerous situation, to vow to refrain from melacha on Chanukah for one or two days; and in that merit they should get saved from harm. This clearly still applies today.
1. The custom is for women to refrain from doing laundry during the first half hour after candle lighting for all eight days of Chanukah.
2. It is permissible for women to cook and write during this period.
3. It is appropriate for women to wear nice clothing and jewelry during Chanukah, within the guidelines of modesty.