Why These Persons?
Within the story the explanation is that these men, when they were alive in the upper world, conducted their affairs with especial attention to fairness. Minos and Rhadamanthys in particular also established the first laws ever used in Greece. Diodorus Siculus says that Rhadamanthys possessed "great justice", and that Minos "ruled wholly in accordance with law and paid the greatest heed to justice."
In Plato's dialogue entitled Minos, the celebrated Athenian philosopher Socrates is deeply engaged in discussion with a friend of his. He alludes to the story about how Minos, every nine years, went up to the Cave of Zeus on Mt Ida on his island of Crete (upon which island both Zeus and Minos were born, as was Rhadamanthys) to consult his father there in order "to learn some things, and to show his knowledge of others that he had learnt from Zeus in the preceding nine years."
Socrates then credits Minos with having somehow legislated moderation in the social consumption of alcohol on Crete, using this as an example of the sort of lawmaking wisdom Minos gained from having been, as he puts it, "a disciple of Zeus". Rhadamanthys, in his turn, was educated in legal matters by Minos, who then appointed him as his attorney general.
Socrates' friend, while acknowledging Rhadamanthys, points out that Minos is notorious for having been "a savage sort of person, harsh and unjust". In his defense, Socrates says that the one mistake which Minos made was going to war against Athens, the work of whose tragic poets is so influential that it could easily destroy the reputation of a just and competent ruler such as Minos certainly was. To reinforce his position, Socrates points out, again, that they in Greece, himself and his fellow Athenians included, were still being governed by the wise laws established by Minos (and Rhadamanthys), who had acquired them directly from Zeus, the god of justice himself.
In the Life of Theseus, Plutarch, who studied philosophy in Athens, seems to be referencing this dialogue when he confirms the idea that it was the poets of the tragedy genre who tarnished Minos' reputation. Curiously, Philostratus puts it in the opposite way in the Life of Apollonius of Tyana, which says that Minos was
a man who exceeded all men in cruelty, and who enslaved with his
navies the inhabitants of continent and islands alike, and yet they
[the poets] honour him by placing in his hand a sceptre of justice and
give him a throne in Haides [Hades] to be umpire of spirits.
The reason for Aiakos [Aeacus] becoming one of these Underworld judges, you have pretty much laid out in your Question. Also Apollodorus tells us that he "was the most pious of men."
Assisting the King
Inside the story, it is mostly incidental that these judges are Zeus's sons. It just so happens that these particular guys were such highly lawful people that they qualified for the task to which they were appointed in the afterlife. Part of the point of the task is for these fellows to help Hades out in his courtroom as he hears the cases of the dead, whereupon decisions are made as to the final destinations of these deceased souls, whether their end is to be good or bad.
Statius' Thebaid seems to suggest that these judges are needed in order to counter the harsh temper of Hades, who is described in Book 8 as "the bloodthirsty king". The same scene portrays the other deities of the Underworld, particularly the river-gods, as likewise participating in the judgements made in this court.
In his 1993 book The Formation of Hell: Death and Retribution in the Ancient and Early Christian Worlds, Alan E. Bernstein relates the myth from Plato's Gorgias which gives us the establishment of the procedure in which certain dead men judge other dead people in the Underworld. The speaker in this dialogue, once again, is Socrates. On p. 56, Bernstein writes:
In the time of Kronos the just were sent to the Isles of the Blessed
to live in happiness and the unjust to the "prison of vengeance and
punishment" called Tartarus (Gorgias 523a). The judges, however,
were living humans unduly influenced by the physical appearance, rank,
and lineage of the people they examined. When Zeus and his brother
took over, Pluto [Hades] denounced this bias.
Zeus therefore reformed the procedure. Humans would no longer have
advance knowledge of their death. They would be tried dead, stripped of all
their worldly accoutrements. The judges, sons of Zeus, would also be
dead, and thus beyond any earthly advantage. From then on Rhadamanthus
tried souls from Asia and Aeacus from Europe. They performed this
function at the Meadow of the Dividing of the Road, where they
separated those going to the Isles of the Blessed from those condemned
to Tartarus (524a). Bearing a golden scepter, Minos presided, ready to
determine doubtful cases.
Family Ties
On a different level, it makes sense for Zeus's offspring to be assisting Hades in his courtroom, considering the close association, in the cults of Hades, between the Lord of the Underworld and the king of the gods, to the extent that Hades was almost considered to be simply a netherworldly version of Zeus, referred to by several epithets which amount to something like "Zeus of the Underworld," a fairly common one being Zeus Khthonios [Chthonius], the "Chthonic Zeus."
The nepotism surrounding those holding the highest positions of power in the universe of Greek mythology is further evidenced by the fact that, as an example, while Zeus is the most important of the Great Twelve Olympian Gods, every single other member of this group is either his sibling or his child (with the exceptions of the versions in which Ares and Aphrodite are not his offspring).
