After Moshe had agreed to go to Pharaoh to beseech him on Klal Yisrael’s behalf, he began traveling to Mitzrayim with his wife Tziporah and their sons – including the newborn. The Torah tells us that they stopped to lodge, and a malach tried to kill Moshe Rabbeinu. Tziporah understood that this was because Moshe had not yet performed a bris milah on their younger son. So she performed the bris milah, and Moshe was saved.
The Gemara in Nedarim 31b explains that the reason that Moshe Rabbeinu had not performed a bris milah on his son was because he was caught in the following dilemma: Hashem had just commanded him to go to Mitzrayim; if he were to perform the bris milah immediately, prior to his departure, he would not be able to depart right away since it would be dangerous for the baby to travel immediately following the bris milah. Also, if he were to wait the necessary time until the baby was fit for travel (three days), he would have prolonged the fulfillment of Hashem’s commandment to go to Mitzrayim.
The Gemara in Avodah Zarah 27a says that there is a machlokes as to whether a woman is allowed to perform a bris milah. The Gemara asks how anyone can say that a woman is unfit to perform a bris milah when the Torah tells us that Tziporah performed a bris milah on her son. The Gemara suggests two answers: 1) that Tziporah did not actually perform the bris milah; rather she arranged for a man to perform it. 2) Alternatively, Tziporah started performing the bris milah and Moshe finished it. (Everyone agrees that a woman may initiate a bris milah. The dispute only regards its completion.)
Even according to the opinion that holds that women are fit to perform a bris milah, the Gemara in Kiddushin 29a derives from a pasuk that women are exempt from the obligation to perform a bris on her son. Tosafos (Avodah Zarah 27a) explains that according to the opinion that a woman is unfit to perform a bris milah, the pasuk is only needed to exempt a woman from the obligation to arrange and ensure that her son has a bris milah performed on him.
The Rishonim in Kiddushin 29a ask the following question: Why did the Gemara deem it necessary to derive from a pasuk that a woman is exempt from the obligation to perform a bris milah on her son? After all, bris milah is a mitzvas assei she’hazman grama (time-sensitive mitzvah), and she should be exempt due to the general rule that women are exempt from all mitzvos assei she’hazman grama.
Tosafos answers that although the mitzvah does not begin until a specific time (the eighth day), since from the time that one can begin performing the mitzvah (from the eighth day forward) it can continuously be performed at all times. It is not considered a mitzvas assei she’hazman grama. Tosafos adds that this answer is only applicable in accordance with the opinion that a bris milah can be performed even at night; otherwise it would not be continuous and thus it would be considered a mitzvas assei she’hazman grama. According to the opinion brought in Yevamos 72 that a bris milah can only be performed during the day, Tosafos still asks why it is necessary to make a drasha to exempt women from the obligation to perform a bris milah on their sons – since in accordance with that view, bris milah is considered a mitzvas assei she’hazman grama.
The Ramban and several other Rishonim write a different reason why the rule that generally exempts women from mitzvas assei she’hazman grama does not apply to this mitzvah. They say that the rule only applies to mitzvos that pertain exclusively to the individual, such as tefillin and lulav. The obligation regarding this mitzvah is to ensure that someone else (her son) has a bris milah performed on him. In such circumstances we do not apply the rule to exempt women from the obligation, and therefore it is necessary to derive it from a pasuk.
The Shulchan Aruch paskins in Yoreh De’ah 264 that a woman is fit to perform a bris milah. There is a discussion as to whether a woman performing a bris milah should recite a berachah, since the pasuk excluded them from this mitzvah.
I think that it is imperative from the abovementioned Rishonim that she would recite the berachah. By a mitzvas assei she’hazman grama, in which women are exempt, most Rishonim opine that a woman who is performing the mitzvah should recite the berachah even though she is exempt. The reason for this is because even though they are exempt from performing the mitzvah, they are nonetheless included in the mitzvah (they are simply not obligated to perform it).