Gemara

A son is in place of his father regarding Yi'ud (a father can be Mekadesh his Amah Ivriyah to his son instead of to himself) and Eved Ivri. (If the father dies, the slave works the remainder of his six years for the son.)

Sotah 49a: Rav Acha bar Yakov raised his daughter's son, Yakov. When Yakov grew up, Rav Acha bar Yakov asked him to get him water to drink.

Yakov: I am not your son! (I need not honor you.)

Bava Basra 91b: "Va'Yasem Kisei l'Em ha'Melech" refers to Rus, the matriarch of the kingship.

Rishonim

Rashi (Bereishis 46:1). Yakov offered Korbanos "lEi'lokei Aviv Yitzchak", and not to lEi'lokei Avi Aviv, for one must honor his father more than his grandfather.

Maharik (30 (44 in Venice edition)): Even if we will say that one should say Kaddish for his grandparent, for grandchildren are like children, they are not totally like children, and a child has more rights than a grandchild. I say that a grandchild has no more rights than a stranger. Kaddish depends on the obligation to honor the Mes. Two brothers receive two shares (of Kaddeshim), for each must honor his parent. We never find that one must honor his grandfather. We find only honoring parents, a step-mother in his father's lifetime, a step-father in his mother's lifetime, and an older brother. All of these are learned from verses. There is no source for a grandchild, just there is a custom to say Kaddish for a grandfather. Grandchildren are like children only regarding Peru u'Rvu, like it says in Bechoros. (We find this in Yevamos 62b - PF.) Mar bar Rav Ashi permits a grandson to testify for his grandfather (Bava Basra 128a)! Even though the Halachah does not follow him, he must be able to resolve the Beraisa that says that grandchildren are like children. The custom is that even a total stranger receives a share, just he gets less than one who says for a parent. We compromise, and he says a third or a quarter as much. All the more so, one who says for a grandfather gets a share! I suggest that he say a third. Regarding being Shali'ach Tzibur, there is no precedence to one whose parent died. It depends on whom the Tzibur desires. It is like a Korban of the Tzibur. One cannot be someone's Shali'ach against his will.

Rebuttal (Teshuvas Rema 118): We find a Mitzvah to honor a grandfather, in Bereishis Rabah (Vayigash 94:5). Yakov offered "lEilokei Aviv Yitzchak", for one must honor his father more than his grandfather. This shows that one must honor his grandfather. All the more so, if the grandfather raised the grandchild, 'one who raises an orphan in his house, it is as if he fathered him' (Sanhedrin 19b), or if he hired a Rebbi for him, he must honor him. One must honor his Rebbi more than his father, for his father brought him only to this world, but his Rebbi brings him to the world to come! Here, his daughter's son says Kaddish for him, just like a regular Avel, as long as she does not object that he says Kaddish in her lifetime. Since it is a Mitzvah to compromise, I suggest like Maharik, that he say a third as much as one who says for a father.

Rebuttal (Bach DH Kosav): It is logical that one must honor his grandfather more than his father-in-law. Regarding testimony it says "Avos Al Banim." It forbids only fathers, but not grandfathers, who are another generation away. We learn Kivud Av v'Em from "Kaved Es Avicha". "Es" includes even step-parents and one's older brother, all the more so a grandfather, who is called a father. Hash-m told Yakov "Ani Hash-m Elokei Avraham Avicha". Also, a grandfather must teach Torah to his grandchild (Kidushin 30a). Surely he is like a father also regarding honor! Also, the reason for Kivud Av is because he is a partner in the formation of the son. This applies also to the grandfather! Tosfos (Yevamos 3a DH mi'Kamei) says that regarding Ervah, a granddaughter is considered a closer relative than a sister. We can say that also honor is unlike testimony.

Note: However, the Isur to marry one's grandmother is only mid'Rabanan!

Maharik (ibid.): Reuven claimed that the custom is that a grandchild does not say for a grandfather. The Or Zaru'a says that a custom must be established through Chachamim. (If not, we ignore it.) Reuven must bring a proof. This is a rare matter, to have two who want to be Shali'ach Tzibur for different Mesim. It happens once every 50 or seven years, so it is not considered an established custom. Perhaps they erred. We rely on customs about matters that occur constantly.

Poskim

Shulchan Aruch (OC 132:2): We say mourner's Kaddish after Aleinu. If there is no orphan in the Beis ha'Keneses, someone missing a parent says it.

Rema (YD 240:24): Some say that one need not honor his grandfather. I disagree. However, one must honor his father more than his grandfather.

Eliyahu Zuta (Sof ha'Sefer, 17:1): The Maharik is correct. The verse did not say 'lEi'lokei Aviv v'Avi Aviv.' This shows that there is no Mitzvah to honor a grandfather. The word '(one must honor) more' is imprecise. Maharik brought a proof from the Gemara (testimony). We do not reject it due to an inference from a Medrash. Another proof is from Sotah 49a. Yakov said to his grandfather 'I am not your son!', i.e. I need not honor you. Presumably, Yakov's father was not there at the time. (I.e. do not say that honoring his father overrode his Chiyuv to honor his grandfather.) Even though Rashi explained 'I need not honor you like a son', he means 'even though I am your daughter's son, I am not like a son, so I need not honor you at all.' According to the Rema, he must honor him in every honor done for a father. The Bach and Shach connote that we force a grandson to honor his grandfather just like we force a son. Do not distinguish between a daughter's son and a son's son. Also, the Medrash discusses a matter in which one must mention and honor his father, and he can honor also his grandfather. One might have thought he should honor also his grandfather, since this pleases his father (who himself needed to attribute the honor to his own father). The Medrash teaches that even so, his Chiyuv to honor his father is greater. However, if his grandfather requests water, or regarding Kaddish, in which there is no Chiyuv to honor his father, there is no Chiyuv to honor the grandfather. However, there is another reason for the grandson to say Kaddish, so that the merit of Kidush Shem Shamayim will come through a descendant of the Mes. One should not do less than the compromise of the Maharik and Rema; the grandson says a third of the Kaddeshim. Beis Din cannot force a grandchild to honor his grandfather, if not due to Tzedakah.

Tana d'Vei Eliyahu (24 DH Kaved v'Lo Signov, cited partially in Shirei Berachah 16): Kivud Av v'Em is written near the Isurim of kidnapping to teach that if a man's children do not honor his father and mother in their old age, it is as if he kidnapped.

Shevus Yakov (2:94): Rashi in Sotah supports the Rema. The Acharonim agree. A daughter's son must honor his grandfather, and all the more so a son's son, for a son stands in place of his father for everything. No one argues about this. Sotah discusses a daughter's son. We find that Shlomo did a great honor for his grandmother Rus, and put a chair for her near himself. If he were not Chayav to honor her, a king may not pardon his honor! I agree that a grandchild should say one Kaddish. Seemingly, his mother can protest. A father can protest that his son not say Kaddish for his mother, because the father's honor comes first (Darchei Moshe 376). (This is letter of the law; the custom is unlike this.) It is rare that a man's son die, and his grandson remains and his daughter-in-law protests, so there is no custom about this. The Rema said that a mother can stop her son from saying Kaddish for her parent. He would not say that she can stop him from saying Kaddish for his father's father, since one must honor his father more than his mother. We find that one does not want his wife to be disgraced (through needing to come to Beis Din) after his death (Kesuvos 74b). All the more so he does not want his father to be disgraced if his grandson will not say Kaddish for him, to fill his father' place! We force the grandson to say Kaddish according to the compromise. If the grandfather taught to him Torah, and he did not leave a son, surely the grandson says Kaddish like other Avelim.

Bi'ur Halachah (Ma'amar Kaddeshim): The Magen Avraham (2) brings from the Rema that if the Mes did not leave a son, the grandson should say Kaddish, but only half as much as other Avelim.

Chasam Sofer: A support (that a grandson can save his grandfather from punishment) is from Yalkut Yeshayah (29, 436) which says that Yakov redeemed Avraham (from the pain of raising children). Also Tosfos (Sanhedrin 19b DH she'Pad'o) explains like this.

Halichos Shlomo 18(50,51, citing ha'Gaon R. S. Z. Auerbach, Ztz"l): If the Mes left no son, each grandson is Shali'ach Tzibur half as often as an Avel, or equally if there is only one grandson. He should not pardon this, for there is no one else to pray for the Mes. He should be Shali'ach Tzibur even when an Avel may not, e.g. on Chol ha'Mo'ed.

Kaf ha'Chayim (16): Sefardi custom allows anyone who wants to say Kaddish to say it, even if several Avelim say it together.

Note: According to letter of the law, only one may say each Kaddish. Nowadays, also most Ashkenazi Batei Kenesiyos allow anyone who wants to say Kaddish to say it. Pischei Teshuvah (YD 376:3) mentions such an enactment.