In his responsa Chavos Yair, Rabbi Bachrach feels that it would be preferable not to say the Kaddish in a public forum, despite the father's dying wish that his daughter do so.
(The fellow had no sons to say Kaddish after him, and was concerned that Kaddish be said by one of his offspring, so he asked that his daughter say the Kaddish).
His feelings were that changes in the form synagogue service traditionally takes will result in people playing with the rules, and a loss in the feelings of deference towards the service.
Additionally, a concern expressed by others was that it would be changing the status  quo of the woman's role in the synagogue.
Indeed, that seems the most common sentiment, and prevailing practice.
However, in the contemporary commentaries (responsa Shvus Yaakov) there exists a consensus that this applies mainly to synagogue service.
Were the prayers to be held at her home, as is often the case during the week of mourning, it would be fine for her to recite the Kaddish in that setting.
Even more startling is the report from a later authority, the Tshuva M'Ahavah, who records the custom in Prague, that after the prayers the elderly men and women in the community used to gather in a room in the synagogue and read T'hillim, psalms.
They would finish the entire book each day.
When they finished the girls who were orphans used to come and say Kaddish before the assembly.
So here we have an actual custom of women saying the Kaddish publicly, although not in the course of the regular daily prayers.
A friend of mine told me that in the community where he lived the older women used to come to the synagogue, and used to say Kaddish too.
This was a strictly orthodox Jewish synagogue, and yet they did say the Kaddish at the end of the daily prayers.
So in conclusion, I can say that it is unusual for women to say Kaddish, and neither I nor most synagogue goers have ever heard Kaddish said by a woman, however there does exist definite historical precedent for girls saying the Kaddish.