This is a difficult one for my Shul, where I am Rosh Hakohol (our Rabbi is ill). We have had a young woman who says Kaddish (quite loud), sometimes in unison with a man downstairs, but occasionally by herself. Some people are upset, others think it’s her right, quoting Rabbi Soloveitchik. She has no brothers.
The question of a woman reciting kaddishhas been discussed by a number of authorities, yet the conclusion remains somewhat unclear, and the answer would depend on the type of shul and congregants involved. If she recites kaddish in unison with a man downstairs, there is room to permit her to continue the practise. She should not say kaddish alone.
Sources: Chavas Ya’ir discusses the question of daughters reciting kaddish, and expresses concern over a woman’s saying kaddish at home (after a learning session), because this could weaken the power of custom. In addition to this, a teshuvah in Teshuvah Me’ahava (vol. 2, no. 239, referring to the same incident in Amsterdam) writes that a woman should not say kaddish in shul. This is also the ruling given by Shevus Yaakov (vol. 2, Orach Chaim 93; cited in Shaarei Teshuvah 132:5), but Sedeh Chemed (Aveilus 160) defers the ruling and states that daughters should not recite kaddish even at home. See also Mateh Efraim, Kaddish 4:8.
However, Rav Henkin ruled that a woman may recite kaddish together with men (Hapardes, Adar 5723, quoted in David Telsner, The Kaddish, p. 301), and this is also the ruling given, as the question mentions, by Rav Soloveichik. This is also implied by Iggros Moshe, Orach Chaim vol. 5, no. 12, sec. 2, who writes that lady mourners would always enter shul to say kaddish. Therefore, provided congregants are happy with it, and in particular after she has already begun to do so (without asking), there is room to permit the woman’s recitation of kaddish, in unison with men. Aside from the issues of tzenius involved and the question of weakening customs, she should not recite kaddishalone because of the problem of her joining the minyan from the ezras nashim (see Aruch Hashulchan, 55:20, who states that somebody in the ezras nashim cannot form part of the minyan downstairs).