Q. I notice that the Yom Kippur poem about the Ten Martyrs mentions Rabbi Ishmael. When did Ishmael cease being a rabbinic name?
A. One of the greatest sages of the Mishnaic period was certainly Rabbi Ishmael the son of Elisha (2nd century). Among other achievements he re-worked earlier material to produce the 13 rules of rabbinic exegesis which are reprinted at the beginning of the Siddur.
There were few rabbis with the name of Ishmael from the time that Islam emerged, presumably because of the significance in Muslim teaching of Ishmael, the son of Abraham.
Yet there were still some Rabbi Ishmaels, such as Ishmael ben Avraham Yitzchak HaKohen of Modena, Italy (d. 1811), who was one of the Jewish leaders to whom Napoleon addressed questions about Jews and Judaism.