In an answer titled "Kaddish for non Jewish Father", you stated that "according to some authorities" it is ok to recite kaddish. A cousin just passed away and I don’t think his mother was Jewish, though his father certainly was. He married a non Jew so his children are not Jewish. Can Kaddish be said for him if it is confirmed his mother was not Jewish? If yes, what is the source.
The kaddish is a prayer of exaltation of the name and glory of HaShem. Its recitation declares our faith in the ultimate revelation of G-d's presence and glory, notwithstanding the fact that He is hidden in the nature of the world. This is not a prayer for the dead, but it does resonate with our faith in the redemption and rebirth of those who have left us. The earliest source for saying kaddish for a non Jew is in the Sefer Chassidim (Rabbi Yehuda haChassid, 12th century Germany) who said that kaddish can be said for a gentile who saved Jews in times of persecution and crisis. Later authorities expanded this ruling for a ger (convert) to say kaddish for his non Jewish parents. Apparently there is no prohibition of saying kaddish for a non Jew, as the kaddish prayer is always acceptable, but the custom seems to have been limited. On the other side of your question, The Chatam Sofer (R. Moshe Sofer, Auato-Hungary, 18th century) ruled that kaddish can be said for a Jewish sinner, even if he has sinned in a way that denies him the world to come. (With one caveat - that the custom is not that only one mourner says kaddish, so as not to deny another the right) My advice to you is that if you wish to say kaddish in honor of your cousin, you may do so, but have in mind the meaning of the kaddish and that you say it with all Jews in mind as well.