The Obligation to Recite the Hallel with a Blessing during the Holiday of Chanukah
In the Shacharit prayer during all eight days of Chanukah, the complete Hallel is recited; a blessing is recited before and after the recitation of the Hallel, as is printed in all Siddurim. Although the Sephardic custom is not to recite a blessing on the recitation of the Hallel on Rosh Chodesh, nevertheless, during the holiday of Chanukah when the recitation of Hallel is an actual obligation and not merely a custom, one must recite a blessing upon doing so like one would before performing other Mitzvot. Clearly, this also applies to both days of Rosh Chodesh Tevet which fall out during Chanukah (this year, 5780, Rosh Chodesh Tevet will fall out this coming Shabbat and Sunday, G-d-willing), in that one must recite a blessing on the recitation of Hallel as one would during the rest of the days of Chanukah. (See Shulchan Aruch, Chapter 683)
The Laws of Women Regarding the Recitation of the Hallel
Regarding the obligation of women to recite Hallel, it would seem that women are completely exempt from reciting the Hallel on Chanukah, for women are exempt from all positive, time-bound Mitzvot (a “positive, time-bound Mitzvah” refers to any Mitzvah which is performed actively, not passively, and is bound by a certain time limit, i.e. that it can only be performed at a certain time; an example of this would be the Mitzvah of Lulav, for this is a Mitzvah that is performed actively and is contingent on a specific time, for the Lulav is taken only during the holiday of Sukkot). It would therefore seem that women should be exempt from reciting Hallel on the holiday of Chanukah as well.
The Reason to Obligate Women to Recite the Hallel
Nevertheless, there is a reason to obligate women to recite Hallel, just as they are obligated to recite Hallel on the night of the Pesach Seder. The reason why they are obligated to recite Hallel on the first night of Pesach is because “they were also included in this miracle,” meaning that since Hallel was instituted to give thanks to Hashem for the miracles He performed for our forefathers and these miracles were performed for women as well, there is therefore no reason to exempt them from reciting the Hallel just as they are obligated to drink the four cups of wine. Regarding the Mitzvah of lighting Chanukah candles as well, the Gemara (Shabbat 23a) states that women are obligated in the Mitzvah of lighting Chanukah candles, for they were also included in the miracle of Chanukah. Based on this, it would seem that women should also be obligated in the recitation of Hallel during the holiday of Chanukah.
The Distinction between the Hallel on the Night of the Pesach Seder and the Hallel During Chanukah
Nevertheless, it seems from the words of the Rambam and other great Rishonim that women are indeed exempt from reciting Hallel during Chanukah. Maran Rabbeinu Ovadia Yosef zt”l explains (in his Chazon Ovadia-Chanukah, page 214) that there is an innate difference between the Hallel recited on the night of Pesach and the Hallel recited during Chanukah, for in principle, our Sages wished to exempt women from the Mitzvah of reciting Hallel since all rabbinic edicts were enacted in a similar fashion to the commandments prescribed to us by Torah law. Thus, since the Torah exempts women from the recitation of the Hallel, our Sages wanted to do the same and exempt them from this Mitzvah. However, on the first night of Pesach, our Sages were compelled to obligate women to recite the Hallel due to the Mitzvah of drinking the four cups of wine since the fourth cup of wine of the Seder night must be drunk alongside the recitation of the Hallel. (The four cups of wine were established to be drunk alongside the four Mitzvot of the Seder night: The first cup with the Mitzvah of Kiddush, the second cup alongside the Mitzvah of reading the Hagaddah, the third cup alongside the Mitzvah of Birkat Hamazon, and the fourth cup alongside the Mitzvah of reciting the Hallel.) He continues discussing this matter at length.
Women Reciting the Blessing on the Hallel
Thus, although women are obligated to light Chanukah candles because they were included in the miracle of Chanukah, they are nevertheless exempt from reciting the Hallel. Therefore, even if a woman would like to act beyond the letter of the law and recite Hallel on Chanukah, she may not recite a blessing before (or after) reciting it since she is not commanded to recite it. The same applies to a woman who takes a Lulav in that she may not recite a blessing before doing so although she receives great reward for doing so as one who is not commanded to perform a given Mitzvah but has done so anyway.
Summary: It is obligatory to recite the entire Hallel, including reciting a blessing on it before and after, every day during the holiday of Chanukah. Women are exempt from reciting the Hallel. Thus, even if they wish to recite it, they may not recite a blessing on its recitation.