The core source is the Shulchan Arukh, Yoreh De'ah 265:11, where the Rema records that it is customary to seek out the honor of serving as sandak, likening the sandak to the kohen who offers the ketoret, and states that custom is not to give two children to one sandak, drawing an analogy to the rule that new kohanim are brought in for each offering of ketoret.
The Levush, Yoreh De'ah 265:11 and the Shakh, Yoreh De'ah 265:22 both explain this restriction the same way: since the sandak is compared to the kohen bringing the ketoret, and the halacha there calls for new kohanim each time, custom likewise directs that each child be given to a different sandak rather than doubling or tripling the same person for one father's children.
The Arukh HaShulchan, Yoreh De'ah 265:34 questions the force of this restriction, noting that a kohen was in fact permitted to bring the ketoret repeatedly and was only excluded so that the honor could be spread among many kohanim; he asks why a father who wishes to give the honor again to the same person should not be allowed to, and observes that in many communities the rav serves as sandak for all the children, comparing the rav in his town to the Kohen Gadol, who was permitted to offer the ketoret whenever he wished.
Rav Eliezer Melamed (Peninei Halakhah, Family 8:11) summarizes the practice as giving the sandak honor preferably to a grandfather or a talmid chacham, and records that the widespread custom is for a father not to honor the same person as sandak twice for his own children, attributing this to the segulah for wealth and blessing associated with the honor and the wish to spread it among worthy people; at the same time he records that others do not refrain from giving the same person sandak honor for two of their children, and that in time of need or by special request parents do give the honor to the same sandak twice.
Arukh HaShulchan, Yoreh De'ah 265:13 addresses a related priority question, ruling that when a child is orphaned of his father and the grandfather is present, the grandfather takes precedence over the sandak in reciting the blessing of hachniso, reasoning that one who is obligated to teach his grandson Torah is all the more obligated to bring him into the brit of Avraham Avinu, though when the father is alive but merely not present, the sandak takes precedence over the grandfather.
Shaarei Halacha uMinhag, Yoreh Deah 97 brings support from the case of Yosef serving as sandak for multiple children of Machir, and on this basis supports the position that a mara de'atra may serve as sandak for more than one child of the same parents, reading the ketoret comparison as applying only to the involvement of other kohanim rather than barring repetition altogether.
Rav Moshe Sternbuch (Teshuvos V'hanhagos 3:293) rules that a father himself may receive the sandakut for his own child, while citing the Rema's preference for selecting the most righteous available mohel and sandak, and noting the particular merit attached to a grandfather serving in that role.
A further practical variation is addressed by Nishmat Shabbat vol 6:3, which records that in Poland the practice was to divide the sandak's role, performing it either all at once or in two stages, by one person or by two, and cites the Ar�up Tzvi as permitting this when the father wants to participate in part of the mitzvah himself, on the principle that a person's agent is considered as himself so that the whole circumcision is still attributed to the father.
The underlying honor accorded the sandak traces back to the Gemara's discussion in Pesachim 7b:2-3 of the brachah recited before circumcision, which frames the father as the one formally commanded in the mitzvah of milah and establishes that he may appoint another to act in his stead, a principle that underlies why the sandak's role is treated as a weighty honor granted by the father rather than an independent entitlement.
