The Gemara (Shabbat 28a) states than an under bar or bat mitzvah katan who lit ner Chanukah did nothing, meaning that other household members cannot fulfill their obligation through him or her: But a woman can certainly light for the household, as women are obligated in ner Chanukah for they too were involved in the miracle of Chanukah.
The Shulchan Aruch (675:3) quotes these rules, and adds: Some say that a katan who reached the age of chinuch, education, can light for others: The reason is that Chanikah is a rabbinic obligation, and a katan is also obligated miderabanan.
The Mishna Brurah (13) says that the Shulchan Aruch omits the view that a katan who reached the age of chinuch may read for a gadol concerning the megilla on Purim, which is also rabbinic : Therefore, [the similar situation] is rejected on Chanukah as well. However, one can distinguish as follows:
A katan's action cannot discharge a gadol's personal obligation, such as megilla, but on chanukah the mitzvah is on the household and as long as the candle has a status of ner chanukah, such as that of an educable katan, it fulfills the obligation of the househols.
Our custom is that every male lights with a bracha to fulfill the mitzva better, as mehadrin. The Mishnah Berurah (9) adds that a woman may make a bracha on her additional ner Chanukah , as she can on any other time-dependent mitzvot aseh such as lulav and sukkah. This statement is very difficult. A woman is obligated in ner Chanukah , and must make a bracha even if she may not make a bracha on sukkah and lulav, as the Shulchan Aruch rules, and is the Sephardic custom.
The answer is that a woman is not required to perform the additional level of mehadrin. But, this too is difficult. Why is she different from a man?
Apparently, women's involvement in the Chanukah miracle obligates them to publicize the miracle. This obligation, known as pirsuma nissa must be obvious to any onlooker. It is obvious that a person is looking at a ner. It is not obvious that halachically one person's lighting fulfills the lighting obligation of all household members. Therefore, women are included only in the obligation of looking at ner Chanukah not lighting the ner. Of course, if a woman lives alone she must light a ner in order to see it.. However, she has no obligation to light as such.
As a result, the idea of mehadrin, that all household members light separately, is incumbent only on men who are required to light. Women need not do so. If they volunteer, they can make a bracha just as when they volunteer to fulfill sukkah and lulav.
In many homes women do not volunteer to light their own mehadring ner- why is that? The Mishna Berurah says (677:16) that a married woman is not included in mehadrin. If she volunteers , her bracha may be a bracha levatala. Therefore, even single women do not volunteer, to avoid either bracha levatala or disappointment over discontinuing lighting ner Chanukah upon their marriage.