Why do some people hold a woman should not say kadish? She's not singing so it shouldn't be Kol Isha, and if she's behind a mechitza it shouldn't be a tzniut problem.
R' Brody discusses the issue on JPost's Ask the Rabbi. He cites many authorities, such as R.Y.D Soloveitchik and R.Y.Henkin, that permitted it and R.M. Feinstein who said it was common practice in Europe. However, he mentions some objections raised:
The permissive position was opposed by many decisors, who argued that (a) women could not participate in this part of the service (Minhat Yitzchak 4:30), (b) the classic mystical sources only spoke of men’s recitation (Mishpatei Uziel OC 3:13), (c) it was immodest in public settings (Aseh Lecha Rav 5:33), or (d) it would ultimately support antinomian trends found in the non- Orthodox movements (Yahel Yisrael 2:90).
It may be that in certain areas the Reform and Conservative movements cause an opposite reaction from Orthodox Jews.
Rav Moshe talks about this in Igros Moshe OC(5) 12:2. He says women did say kaddish but the tshuvah is more focused on a mechitza for one or two women .
מ"מ יש לחוש שע"י כך יחלשו כח המנהגים של בני ישראל שג"כ תורה הם ויהיה כל אחד בונה במה לעצמו ע"פ סברתו ומחזי מילי דרבנן כחוכא ואטלול' ויבואו לזלזל בו
Nevertheless there is concern that through this the strength of customs of the Jews – which are also Torah – will be weakened, and each one will construct a bamah for himself according to his reasoning, and rabbinic matters will appear like a mockery and a jest, and they will come to debase it.
Quoting from Mourning in Halacha:
Mishmeres Shalom(os kuf 51) cites the Bais Lechem Yehudah (376:5) who states in the name of Sefer Chassidim there is no halachic basis for a daughter’s reciting Kaddish and (such a practice) is ridiculous nonsense. The same ruling is cited by the Sdei Chemed ( Aveilus 160) citing many Acharomin.
Chavos Yair according to basic of the law she should recite Kaddish because it brings satisfaction to the soul. Nevertheless, there is danger that this practice would weaken the customs of the Jewish people.
Interestingly the Mateh Efrayim ( Dinei Kaddish Yasom 4:8) says that if a father had commanded his daughter to say Kaddish the command should not be obeyed and one should protest against such a practice even in a prayer service in the daughter’s home and in a synagogue. Instead, she should listen to someone who is saying the Kaddish and answer amein etc.
Some feel that women should not recite kaddish. For example, R. Shimon Frankfurter said that "a woman's voice (קול אישה) (when singing) is lewdness." But when she says the Mourner's Kaddish, there is no lewdness or licentiousness, therefore, making it impossible for a man to be triggered. And even if the male was triggered, the issue would not be with the female. R. Frankfurter further states that women sanctifying the name of G-d violates kavod tzibur, but the idea is only limited to Torah readings. Lastly, they say that saying Kaddish is "inappropriate" for women. However, R. Moshe Feinstein, R. Yosef Eliyahu Henkin, R. Yosef Soltoveitchik, R. Yosef Dov Soltoveitchik said that women could perform the practice. The following are some reasons why I think women can and should recite Kaddish.
Although there is no reference to death in the kaddish prayer, and although the oldest text of Kaddish, written in Aramaic, dates almost back to a thousand years, from a siddur of R. Amram Gaon, and to say nothing that Maimonides does not even mention this practice, confirming to the view that it was developed; it is nevertheless, the process of sanctifying G-d's name, a קדוש השם, which is actually a mitzvah! Actually, its more than a mitzvah, its incumbent to both genders and thereby exempts mitzvot aseh shehazman grama since sanctifying the name of G-d isn't a time-bound mitzvah. In other words, to restrict women from saying Kaddish is actually a violation of לפני עבר and; thus is a violation of the Torah!
Besides, it would be a Kiddush Hashem and in the presence of a minyan. Besides, Rabbi Soloveitchik ruled that women could recite the kaddish, thus respecting departed loved ones. Not to mention that the Mourner's Kaddish is a minhag (custom), and not biblically mandated per se, means that women can take part in saying the Kaddish.