There are many problems with the thread title. First, ancient Athens as a city may never have had more than 40k people; the size of the whole state can only be guessed, though 250k including enslaved people is a reasonable estimate. Second, the people you list didn't just pop out of the ground in 500 BC; Solon and Pythagoras lived generations before that date while Archimedes died three centuries after. If you consider that these people actually existed across a span of 400 years, the supposed achievement already appears much less impressive. Thirdly, as you say, many of them were not Athenian (this also includes Aristotle, and arguably the half-Thracians Perikles and Thucydides). The Greek world of the Classical period represents a share of the world population that was, at its highest estimate, comparable to the US today; it is not surprising if some of the millions of ancient Greeks managed to leave a mark.

But the bigger problem is, of course, the notion of a "Greek miracle" itself, which you define even more narrowly here as an Athenian miracle. Some scholars still adhere to the idea that something special must have happened in Athens during the Classical period, which made the city a uniquely fertile ground of intellectual development (though I don't think anyone would include Pythagoras and Archimedes in this narrative, as they really don't fit well either chronologically or as examples of Athenian achievement). Most of these scholars will argue that this special thing was Athenian democracy. Supposedly, the political culture of equality, free speech, deliberation and persuasion spawned a culture of intellectual freedom and innovation that has few rivals in history. Others have also stressed that Athens was an imperial city during the fifth century, attracting skilled labour, artists and philosophers because of the relatively greater amount of money and opportunities available. Imperial Athens was full of resident foreigners from all over the Mediterranean who freely exchanged knowledge, skills and ideas; even the Greeks at the time recognised that it was immigration that fuelled Athenian greatness. If you want a plain answer to the question, this would be the one offered by most people who believe in a "Greek miracle". Fifth-century Athens prided itself on its dynamism, meritocracy, and innovative energy, as Perikles lays out in his funeral oration as reported by Thucydides. New forms of literature and philosophy are some of the products of that culture.

But we should read this more critically. Of course the Athenians are going to say that their empire and democracy made them into a shining beacon of equality, opportunity and human achievement; it was essential to present themselves as such to deflect accusations that they were enslaving the rest of the Greek world. Their ability to patronise artists and develop all forms of art and literature, after all, was paid for by their subject allies. The Athenians were the first to levy tribute on other Greeks, and any state that tried to break away from this obligation was ruthlessly punished. For all its supposed brilliance, Imperial Athens was, according to Thucydides, hated by all; the Spartans had an easy time justifying the Peloponnesian War as a crusade to "liberate the Greeks".

And once we realise that the picture of Athens as the "School of Greece" (again, Perikles) is a product of their relative abundance of wealth as well as their self-serving propaganda, we can also start to reassess the merits of those supposed paragons of brilliance. The problem here is not so much that Athens didn't produce geniuses, as it is the fact that we still define "geniuses" by the Athenian standard. When we assume that intellectual greatness comes in the form of Great Statesmen and Great Philosophers, we are perpetuating a notion born out of what Athens was like at this time. The influence of ancient Greece on our understanding of intellectual achievement is such that we still don't have any other paradigm to define or measure it. In a sense, when we say that a particular person or group is brilliant, we are still saying little more than that they are like the ancient Greeks. And unsurprisingly no one is as much like the ancient Greeks as the ancient Greeks themselves. But what did those Greeks really achieve, that they still provide the benchmark against which we compare others (and even ourselves)?

It is interesting to note here that a lot of supposedly great Greek ideas don't actually have a "father" (let alone a mother). We don't know who really invented tragedy, or democracy, or Pythagoras' theorem (which is many centuries older than Pythagoras, as Near Eastern sources clearly demonstrate). In case where we have a clearer idea, like historical writing or rhetoric, it is clear that there was no single brilliant invention but rather an incremental development and steady cross-fertilisation of a number of different ideas of which the origins are beyond recovery. Yes, many of these things developed in the Greek world, often some time between 600 and 300 BC - but they weren't the miraculous inventions of a single genius. Rather they are the product of an interconnected world in which many new ideas were finding a place and an audience. Similar things were going on elsewhere at the time - especially in the East - and many scholars have studied their influence on Greek thought. To a large extent the fact that it is Greek ideas that persist to our time (and not, say, Phoenician or Babylonian) has more to do with accidents of survival and accessibility than it does with a uniquely productive intellectual culture.

Meanwhile the figures that have been idealised as perfect geniuses for thousands of years - figures like Perikles or Sokrates - are often only tenuously associated with concrete ideas. What good did it do anyone that Perikles was in charge of Athens for a few decades in the fifth century, when we struggle to identify a single reform he enacted? What good did it do anyone that Sokrates hung around Athenian public spaces asking questions, given that it is now impossible for us to tell whether he ever had any ideas of his own? At best, such figures were part of the environment that allowed others to develop their own ideas, but even the merit of those ideas can be questioned. Aristophanes' mockery of Sokrates and others like him as parasites and charlatans tells us a great deal about how much ordinary people cared or were affected by the supposedly great discoveries of Athenian philosophers. How many lives did these men actually affect while they were alive?

Instead, we should be aware that when we think of Athens as a society of geniuses we are simply repeating the narrow, self-serving, uncritical propagandistic picture they liked to broadcast. At the very least we are forgetting to ask whether Plato and Thucydides can be thought of as representative ancient Greeks, when they represented only the absolute upper crust of society - the richest of the male citizens. We should remember that whatever went on in their circles would leave the vast majority of their contemporaries totally unaffected, if not outright invoking the wrath of the common people for the frivolousness and practical irrelevance of their interests. What juice were they drinking? Apart from the ever-bitter juices of Empire and privilege, they drank from the cups of the narrowest elite priding itself in the novelty of its latest cleverness, within a highly interconnected intellectual world, which, yes, did briefly produce some quite exceptional minds.

I strongly disagree with the main thesis /u/Iphikrates's answer (although certainly not with all the points made). In the spirit of friendly scholarly debate, I thought I would leave an answer putting forth an alternative view. I believe I am justified in making this a new post because I am not simply responding to /u/Iphikrates's answer (although I do refer to it) as I am trying to answer /u/RusticBohemian's question itself.

It is right to note that /u/RusticBohemian's question is a bit confused, for some of the famous names mentioned are not Athenian. However, as the original question did ask specifically about Athens, I will focus on that city.

I Why do we know these names?

Strictly speaking, the Athenians asked about were Socrates, Plato, Aristophanes, Xenophon, Thucydides, Solon, and Pericles. However, I will also include Aristotle in the list, for, although Macedonian, he was a student of Plato who spent much time in Athens; he was certainly a major player in Athenian intellectual life.

What /u/Iphikrates leaves out is the entire reason that these names are so famous today. While they are all mentioned by various works in the extant Greek literature, five of them (Plato, Aristophanes, Xenophon, Sophocles, and Thucydides) are also authors of some of the most famous of this literature. Usually, we classify Plato's and Aristotle's works as philosophy (tradition handed down 35 Platonic dialogues as well as 13 Platonic letters as well as numerous Aristotelean treatises; whether these are all authentic is a matter of scholarly debate), Aristophanes as comedy (tradition handed down 11 comedies), Sophocles as tragedy (tradition handed down 7 tragedies), and Thucydides as history (just one work, on the Peloponnesian War between Athens and Sparta). Xenophon is harder to so simply categorize, as he wrote what we would call history (such as his Hellenica) as well as philosophy (such as his Memorabilia). In any event, we have 15 of his works (but, like Plato and Aristotle, whether all of these are authentic is debated). Had these works—not only their "ideas"—not survived, they would not still be as famous as they are.

/u/Iphikrates is right to point out that part of the reason we have these works today is "accidents of survival and accessibility," but that is not the whole picture. These works were also intentionally chosen to be preserved by scholars at the library of Alexandria. As Knox explains:

In the final, desperate centuries of classical civilization, the years of civil wars and massive foreign invasions, the vast bulk of ancient Greek literature, including, to our everlasting lost, most of the work of the nine lyrics poets, vanished. The last copies disintegrated or were burned in the sack of the cities that housed them. Only those works transferred to the more durable (and expensive) material of parchment could survive, and in what was now a Christian world the pagan authors preserved were those thought necessary for the schooling of the young—a restricted version of the original canon: Homer, Hesiod, Herodotus, Thucydides; seven tragedies each for Aeschylus and Sophocles; ten for Euripides; eleven comedies of Aristophaes; and, partly no doubt because of the influence of Neoplatonic philosophy on the Christian theologians, all of Plato, and much of his successor Aristotle.

(Knox, Bernard M. W., The Oldest Dead White European Males, 14–15).

Our question therefore becomes why these works were preserved. Why were these works specifically chosen to be worthy of the time and effort it would take to transfer them to parchment? Would they have been if, simply "apart from the ever-bitter juices of Empire and privilege, [their authors] drank from the cups of the narrowest elite priding itself in the novelty of its latest cleverness"? Could it have been at least believed that they drank from the cup of truth concerning human nature or the human experience as such?

Maybe, maybe not. Knox does point out that those who chose to preserve these works had motives in mind other than the pursuit of truth (yet also other than "the Athenian elite"): they liked Plato because Neoplatonism went well with Christian theology. But, of course, these Alexandrian libraries are not the only individuals who preserved these works: countless human beings had to come to the conclusion, again and again, over the years, that these works were worth preserving. Why is this? For whatever reason, we have the works today, and so we can try to judge their worth for ourselves. If you are interested, I highly encourage you to do so. You may discover that these works are inherently limited by the elite prejudices of their authors. But you also may find that they do point to an understanding of what it means to be a human being unlike any other literature, either throughout history or today. If the latter is the case, then the notion of Athenian genius would not simply be a myth perpetuated by Pericles' propaganda.