The Torah tells us that Yosef's great- grandchildren, specifically, the sons of Machir, who was the son of Yosef's son Menasheh, were born "על ברכי יוסף," which literally means "on Yosef's knees" (בראשית נ:כ"ג). The Targum Yonasan Ben Uziel (שם) understands this expression as a reference to the act of circumcision, the Bris Milah, teaching, as suggested by one opinion cited in the commentary there (פי' יונתן שם), that Yosef served as the Sandak at the Bris Milah of each of these great-grandsons. The Yalkut Shimoni (חלק ב' רמז תשכ"ג), commenting on a Posuk in Tehillim (ל"ה:י'), presents a list of various different parts of the body, and describes certain Mitzvos which can be done with each one of them, and notes that with one's knees, one can serve as a Sandak for baby boys during their Bris Milah. The Ohr Zarua (חלק ב' סימן ק"ז) quotes this same idea from the Midrash Shochar Tov on Tehillim (שם), as does the Maharil (ספר מהרי"ל, ריש הלכות מילה), although the current standard editions of the Midrash Shochar Tov do not seem to record it, and they explain that the job of the Sandak is to hold the baby on his lap while the Milah takes place; the Sefer Rokeiach (סימן ק"ח) quotes this Midrash as well, and then cites another Midrash which indicates that Hashem Himself was the Sandak at the Bris Milah of Avraham Avinu. The Aruch HaShulchan (יורה דעה סימן רס"ה סוף סעיף ל"ד) notes that the word Sandak itself (סנדק) is not originally a Hebrew word, and is defined as a word which in Greek and Latin means a godfather; the Avudraham (שער ט', הל' מילה וברכותיה, עמוד שנ"א) defines it this way as well, adding, as do others, including the Ohr Zarua (שם) and the Maharil (שם), that the person designated as the Sandak is considered a בעל ברית, one of the important participants in the Mitzvah of Bris Milah.
The Ramo writes in the Shulchan Aruch (יו"ד שם סעיף י"א), based on a comment of the Hagahos Maimoniyos (לפרק ג' מהל' מילה הלכה ט', אות ה'), who also cites the aforementioned Midrash Shochar Tov (שם), that the custom is to actively pursue this job of serving as the Sandak at a Bris. It appears from a statement of the Shach (יו"ד סימן רמ"ט ס"ק ג') that people would actually pay money in order to get this job, and that money set aside for Tzedakah as part of one's obligation to give Maaser Kesafim (one tenth of one's income) as charity may be used for this purpose; the Chofetz Chaim, in his Sefer Ahavas Chessed, (חלק ב', ענין מעשר כספים, פרק י"ט אות ב') notes that this money is given to the father of the child about to have his Bris Milah, and that if the father is a poor man, one may certainly use Maaser Kesafim money in order to "buy" this right to serve as the Sandak at the Bris. The Ramo (שם) also rules that a woman should not serve as the Sandak if there is a man who can do so, because this would reflect a lack of modesty, but a woman can bring the baby to (the entrance of) the Shul, where the man who is the Sandak takes the baby from her. The Aruch HaShulchan (שם סימן רס"ד סעיף ל"ה) notes that this is not exactly our practice today, as a woman does not directly assist the Sandak at all; rather, there is a separate honor that when the baby is brought to the place where the Milah will be done, women bring the baby to the door of the room, where one woman then stands while holding the baby on a pillow or a blanket, and a man then takes the baby from her and brings him forward to where the Bris will take place. The person who brings the baby to this place where the Bris will be done is called a "Kvatter" (קוואט"ר).
The Maharil cited above (שם) writes that the Mitzvah performed by the Sandak at the Bris is greater even than that performed by the Mohel himself, and that for this reason, the Sandak takes precedence over the Mohel in terms of an entitlement to be called for an Aliyah to the Torah (on the day of the Bris); the Ramo (שם סעיף י"א) rules accordingly. The reason suggested for this by the Maharil (שם) is that the legs of the Sandak are compared to a Mizbeiach, and they are viewed as though the קטורת, the incense offering, is being offered to Hashem on them; the Ramo (שם) writes simply that every Sandak is considered as though he is offering the קטורת. The Sefer Otzar HaBris (חלק א' פרק י"ד הערה ו') suggests that this idea may be based on a statement in the Midrash in Bereishis Rabbah (פרשה מ"ז סימן ט') and in Shir HaShirim Rabbah (פרשה ד' סימן י"ד) which draws a connection between the Bris Milah and the קטורת, or perhaps on a statement in the Midrash Tanchuma on Parshas VaYeira (סוף אות ב') which indicates that Hashem prefers that which results from a Bris Milah to the spices used for the קטורת offering. It should also be pointed out that the Midrash in Bemidbar Rabbah (פרשה י"ד סימן כ"ד), when discussing a Posuk (במדבר ז':נ"ו) which describes a certain קטורת offering, connects the offering to the Mitzvah of Bris Milah.
Based upon this connection between bringing the קטורת offering and serving as a Sandak at a Bris Milah, the Maharil (שם) quotes from Rabbeinu Peretz that the practice is for a father not to have the same person serve as the Sandak for more than one son, but rather to select a different Sandak for each son. This ruling is based on the statement of the Mishnah in Yoma (דף כ"ו.) that only a Kohein who had never brought the קטורת offering would be invited to bring it on a given day; the Gemara (שם) then explains that a Kohein who has once brought the קטורת offering would not ever bring it again, and the reason, based on Pesukim later in the Torah (דברים ל"ג:י'-י"א), is that offering the קטורת can bring one wealth as a reward, and therefore, as noted by Rashi (שם בד"ה מפני), nobody was given the opportunity more than once. The Maharil (שם) thus says that just as a different Kohein was always given this chance to become wealthy through the קטורת offering, so too a different person should be given the chance to serve as a Sandak for each child that one has. This ruling is accepted and quoted by the Ramo cited above (יו"ד שם); the Shach (שם ס"ק כ"ב) implies that this restriction upon being a Sandak for more than one child applies only if the children have the same father, but one may, apparently, serve as a Sandak as often as one likes as long as the children are all from different families. Rav Ovadyah Yosef (שו"ת יחוה דעת חלק ג' סימן ע"ז) cites one authority that writes that one should not serve as a Sandak more than once in his lifetime, even for children from different families, but he concludes (שם) that this is not the majority view.
The Vilna Gaon, however (ביאור הגר"א ליו"ד שם ס"ק מ"ו), in fact challenges this entire idea of restricting one from serving as a Sandak for more than one child of the same family based on the connection to the קטורת offering, because he believes that if this connection were really meant to be taken so literally, it would indeed be prohibited for one to serve as a Sandak more than once in his life, not only if the children are from the same family, but no matter who the children are. The Noda BeYehudah (שו"ת נודע ביהודה מהדורא קמא חלק יו"ד סימן פ"ו) also questions this entire connection between serving as a Sandak and offering the קטורת, noting that there isn't even a hint in the Gemara to this idea of limiting one's capacity to serve as a Sandak, and demonstrating that the comparison to the קטורת offering (which was brought on the inner Mizbeiach) is difficult, and that it would have made more sense to compare serving as a Sandak to offering a (standard) Korban on the outer Mizbeiach; these offerings could certainly be brought by the same Kohein more than once. Interestingly, the Vilna Gaon (שם ס"ק מ"ה) does actually compare the Sandak's Mitzvah to a Korban, an idea also found in the Pirkei DeRabbi Eliezer (פרק כ"ט, ועיין שם בביאור הרד"ל אות מ"א) and in the Yalkut Shimoni (חלק א' רמז פ"א), as well as in the commentary of Rabbeinu Bechaya on the Torah (בראשית י"ז:י"ג).
The Noda BeYehudah (שם) does go on to defend the comparison to the קטורת offering, citing one of the Midrashic sources quoted above, but he does not believe that this comparison means that serving as a Sandak likewise brings one wealth and that one should therefore not do it more than once, and he tries to prove his position based on the aforementioned Gemara in Yoma (שם). He then concludes by saying (שם) that while it should be up to the father of the baby who is about to have the Milah to ask whomever he wishes to serve as the Sandak (because this situation is really unlike the situation regarding offering the קטורת, where all Kohanim have an equal share in the Mitzvah, and it would thus be unfair for one Kohein to do it more than once while another may never get a chance), there is a Gemara in Eiruvin (דף ס"ג.) which indicates that one should not give all his Matnos Kehunah (that is, those gifts which every Jew must give to a Kohein) to the same Kohein, and therefore, perhaps, one should not always ask the same individual to serve as the Sandak. This approach explains why the prohibition against being a Sandak for more than one child applies only regarding children from the same family. The Noda BeYehudah (שם) stresses again, though, that this whole matter is not rooted in the Gemara, and he notes that there is no strict Halacha or fixed Minhag about this, and that indeed there are many communities that do not observe this practice at all.
The Vilna Gaon (שם ס"ק מ"ו) also questions this practice by showing that the premise is problematical, because there is no evidence that one becomes wealthy because of serving as a Sandak, and he concludes (שם) that the custom not to serve as a Sandak for more than one child in the same family is actually based on a statement to this effect in the will of Rabbeinu Yehudah HaChassid (צוואת רבינו יהודה החסיד סימן ל"ה). The Aruch HaShulchan (שם סעיף ל"ד) also concludes, citing this same ruling of Rabbeinu Yehudah HaChassid (שם), that this practice is based on certain hidden reasons, while the comparison to the קטורת was just an attempt to find a hint to it, and he rules that we should certainly observe it, as indicated by the Ramo (שם). Interestingly, he explains (שם סעיף ל"ה) that the assistance provided by the "Kvatter," referred to above, is also connected to the קטורת, and he suggests that the very word "Kvatter" (קוואט"ר) is somehow a permutation of the word קטורת. As noted by the Pischei Teshuvah (שם ס"ק י"ד), the Shaarei Teshuvah elsewhere (אורח חיים סימן תקנ"א ס"ק ג') also questions the practice of avoiding serving as the Sandak more than once, since the Sandak, as quoted above from the Maharil (שם), is compared to the Mizbeiach, not to the person offering the קטורת, but he too ultimately justifies this idea based on another Gemara in Yoma (דף כ"ה.).
The Chasam Sofer (שו"ת חתם סופר חלק או"ח סימן קנ"ח) also discusses this entire issue at length, and he justifies the practice of not being a Sandak for more than one child in the same family. He writes (םש) that the observation of the Vilna Gaon (םש) that people do not become wealthy after having served as a Sandak is inconclusive, because other circumstances and deeds may prevent one from receiving that reward. He also documents possible sources for the entire comparison between the קטורת offering and the Sandak, and he responds to some of the issues raised by the Noda BeYehudah (םש); he asserts that this practice should certainly be observed today, though he notes that the Rav or a great Talmid Chochom in the city can be honored as the Sandak more than once even within the same family, because the Kohein Gadol was allowed to offer the קטורת more than once, and the Rav is thus like the Kohein Gadol in this respect. Rav Ovadyah Yosef, in his aforementioned Teshuvah (םש), quotes a view that one may ask a relative to serve as a Sandak for more than one of his children, since this entire restriction does not apply to relatives; the Chacham Tzvi (שו"ת חכם צבי סימן ע') implies that a relative takes precedence over a Talmid Chochom for this honor to serve as a Sandak. Nevertheless, it is clear from the Maharil (םש) and the Ohr Zarua (םש) cited above that one should be sure to find a completely worthy person to serve as the Sandak, so that Eliyahu HaNavi will accompany him at the Bris; the Ramo (שם סימן רס"ד סעיף א') rules accordingly. As for serving as the Sandak for more than one child in the same family, it appears that most authorities hold that one should avoid it, but it would also seem that one may also follow the practice of his community.