A minor whose mother or father died says Kaddish for his parent though he has not yet reached the age of mitzvot (bar mitzvah). Mourner’s Kaddish was instituted for that purpose, since a child cannot lead the prayer service in order to facilitate the ascent of his parent’s soul. Therefore, the Chachamim instituted Kaddish (Rama, Yoreh De’ah 376:4). Even if the minor has not yet reached the age of understanding, the Kaddish is read to him; the young orphan repeats the Kaddish word for word, and the congregation answers Amen. (According to the Ari, it is important to be strict and have an adult say the Mourner’s Kaddish with the minor, because that Kaddish is included in the twelve Kaddishim that are necessary to hear every day. See further in this book 23:10.)
In the case of a man or woman who was not privileged to leave behind a son, a God-fearing grandson may say Kaddish for him or her the whole year. This is possible even if the deceased has a son who is not God-fearing and does not go to synagogue to recite Kaddish. A grandson from a son takes precedence over a grandson from a daughter. If the deceased does not have a grandson yet, but has a son-in-law, the son-in-law should say Kaddish. The grandson and son-in-law are permitted to say Kaddish only when one of their own parents is deceased, or if their parents agree to his reciting Kaddish while they are living. However, someone whose parents are strict that he not say Kaddish while they are both alive may not recite the Kaddish for his grandparent or in-law.
When the deceased does not have a son, grandson, or son-in-law, the father of the deceased recites the Kaddish. If his father is also dead, the brother or nephew says Kaddish.
When none of these relatives can say Kaddish for the deceased, part of the inheritance money should be used to hire a God-fearing person to recite Kaddish for him. It is good to hire someone who is engrossed in Torah. If there is someone in the family who is occupied with Torah study, he takes precedence over a stranger. The monetary compensation for the Kaddish is important in order to ensure the fulfillment of its recital. Furthermore, by employing someone who is involved in Torah or a poor person who has children to support, the deceased will acquire more merit.
If a minor who already reached the age of understanding passes away, his father must say Kaddish for him or her (Pitchei Teshuvah, Yoreh De’ah 376:3). There are those who have the custom to say Kaddish even for a small baby (see Pnei Baruch 34:30).
One who hires himself out to recite Kaddish may do so for a number of people, on condition that he ends up saying at least one Kaddish a day for each of the deceased (Igrot Moshe, Yoreh De’ah, part 1, 254, and see Pnei Baruch 34:23-28).
It is proper for an adopted son to say Kaddish for his adoptive parents. All the more so, if the adoptive parent does not have another son, it is a mitzvah for the adopted son to say Kaddish (Yalkut Yosef, part 7, 23:13). It is also good for a righteous convert (ger tzedek) to say Kaddish for his gentile parents (Yalkut Yosef, part 7, 23:14, and Piskei Teshuvot 132:20).
. Even when the deceased has a daughter, a man should be hired to say Kaddish. However, throughout the generations, there were places where the daughter would say Kaddish, either in her home or in the room adjacent to the synagogue, if there were no living son. Additionally, there are those who ruled that if the daughter is younger than 12 years of age, she should recite Kaddish in the synagogue. Nevertheless, the accepted minhag (custom) is that women do not say Kaddish. The Chavot Yair, responsa 222 states that we must object to women saying Kaddish so as not to undermine the power of minhagim, as also written in Yalkut Yosef, part 7, 23:11, Pnei Baruch 34:20, and Piskei Teshuvot 132:33.