Socrates had three sons according to Plato's dialogues, as you mention. In Apology 34d he says:

I have a family too, and indeed sons, men of Athens, three of them, one already a teenager [μειράκιον meirákion] and two who are children [παιδία paidía].

We know that the teenager was named Lamprocles, and he actually appears briefly in Xenophon's Memorabilia. According to Xenophon, Lamprocles had just had a fight with his mother. Socrates told his son that he should feel grateful for all his mother had done to raise him, resulting in this exchange (translation from Delphi Classics):

To this [Socrates's explanation of all that his mother had done for him] the young man replied: "Nay, but even if she has done all this and far more than this, no one could put up with her vile temper."

"Which, think you," asked Socrates, "is the harder to bear, a wild beast's brutality or a mother's?"

"I should say a mother's, when she is like mine."

After a few more back-and-forths, the conversation ends with Socrates asking Lamprocles whether he cares about strangers' opinion of him. When Lamprocles says he does, Socrates responds:

"And yet, when you are resolved to cultivate these, you don't think courtesy is due to your mother, who loves you more than all? [...] Should men suppose you to be ungrateful to your parents, none would think you would be grateful for any kindness he might show you."

This is the only real insight we have into Lamprocles, and he doesn't make a good showing.

According to the Roman-era biographer Diogenes Laërtius, the two younger sons were named Sophroniscus and Menexenus, and they were born to Myrto and not the (in)famous Xanthippe. Menexenus, son of Socrates, is very likely unconnected to the young man Menexenus who appears in the [Platonic dialogue named after him](https://en.wikipedia.org/wiki/Menexenus_(dialogue)), although Classicist Lesley Dean-Jones has tried to argue otherwise.

Now, why are they virtually unknown? Well, Aristotle's Rhetoric might hold the answer. In Rhetoric 2.15, Aristotle says (emphasis mine):

Highly gifted families often degenerate into maniacs, as, for example, the descendants of Alcibiades and the elder Dionysius; those that are stable into fools and dullards, like the descendants of Cimon, Pericles, and Socrates.

Presumably none of the three sons were particularly accomplished.