For my part, there are two approaches, one of which has already been discussed. The one everyone's talked about is "top-down," where you look at the high-level themes or at literary, mythic, religious, etc. source material and draw forth stories that way. The other approach is "bottom-up," and it's...well, a lot more speculative, less likely to be instananeously grabbing, but more likely to be very natural and well-suited to the specific thing you've created. Actual D&D races have had both.
Since it's sort of my traditional topic in this sort of thing, I give the example of dragonborn, viewed through each lens.
From a top-down perspective, dragonborn are pretty obviously reptile-like humanoids. That means you can potentially tap into a whole host of lizardy-boi stories across sword and sorcery, myth, legend, you name it. But because they have a connection to dragons, that kicks things up a notch--and also enables some really interesting connections specifically to several Greek myths. See, there are several Greek myths that involve something surprisingly similar to dragonborn (minus the breath weapon, since that's a more modern invention WRT dragons). You have the obvious direct parallels like the Spartoi, the warriors literally sown from dragon's teeth, or the ophiogenees, literally the "serpent-born" who were the descendants of the Mysian Dragon or Drakon Ophiogeneikos (literally "the dragon of the serpent-born") and Halia, a woman who had come to worship at the sacred grove of Artemis that the Mysian Dragon guarded.
But there are other myths too, like the story of Erikhthonios (often rendered "Erichthonius" or "Erechtheus," as these were different names but the earliest attestations use them interchangeably): Hephaestus came to desire Athena after Aphrodite had spurned him, but Athena rejected his advances; he tried to have his way with her but she fought him off, and wiped his...seed...off of her thigh with a bit of wool, causing a child to be born from the earth--but this child was either "a serpent intwined around the babe" or actually himself "half-human, half-snake," to the point that some myths explicitly call him a "drakon." He's also considered Athena's son (since she raised him and...participated...in his conception) even though she remained forever a virgin--a lovely example of mythic contradictions resolved. There's also multiple references to drakaina, female dragons or half-human half-dragon women, some of whom produce lineages of kings, e.g. the Scythian drakaina, who struck a deal with Hercules, that he become the father of her descendants.
So there's actually an awful lot of heroes, tribes, and royal lineages that could have been drakogenes, "dragon-born," in Greek myth. Perhaps dragonborn have the quirk that they are always born from an actual dragon's teeth (maybe dragons are like sharks in this setting, so loose dragon teeth are more common than one might expect), adult and fully formed. That creates one hell of a distinct story: what's it like for a culture where there are no children, no mothers, and no need for marriage or partnership except as a choice? Alternatively, perhaps dragonborn in this setting take cultural cues from the culture that the Greeks called "Scythians," a broadly successful culture from around 900 BC to ~500 BC (though remnants lingered well into the first millennium AD). The Scythians were noted for their excellence in war (Herodotus claimed the Scythians could not be stopped by anything less than an alliance of several states), rich artistic tradition particularly in metalwork (some of which survives to this day), and participation in the early Silk Road. That's a lot of cultural toys to play with, just gotta find a path that works.
Alternatively, you can build bottom-up. Dragonborn (at least in 4e...) have a bonus to History skill, high Charisma and either Strength or Constitution (meaning, overall high fortitude no matter what), and tend to heal from injuries more quickly. They also need less time to develop: female dragonborn are only "pregnant" (carrying an egg) relatively briefly compared to human pregnancies, then the fetus finishes maturing inside the egg in a hatchery of some kind. (In this regard, they are like hairless monotremes, or as I like to phrase it, "reptile-like mammals" rather than "mammal-like reptiles.") They mature extremely quickly, reaching the equivalent of a 10-year-old human child at merely 3 years, finishing their growth spurt by about 12-13, and being fully physically adult by 15; their maximum age is approximately the same as a human's. Most adult dragonborn have a breath weapon (though some have dragonfear instead), and although they are not dragons, they have a blend of draconic pride and more mortal-like desire for social companionship, which tends to drive them toward pursuit of excellence and clannish/reputation-heavy behavior. Their diet is heavier on protein than a human's, but otherwise the same, and they can tolerate the same temperature ranges as humans do (thanks to their wide mouths, which allow substantial heat transfer).
From the above, we can do all sorts of stuff. Dragonborn are bright fires, stars that flare hot rather than fading away--they may unfortunately embody the "live fast, die young" mentality, just with "glorious" replacing "fast." Their early lives are extremely brisk; most dragonborn are full adults a year or more before human high school students begin thinking about their college prospects, and their infancy/childhood is incredibly short, cramming into three years what humans take 10 to work through. This is likely to lead to a society that looks a lot more like what seems to occur so often in anime: prodigies of (what we would consider) shockingly young age yet superlative skill nonetheless; to them, Alexander the Great would have been something of a late bloomer, not the ridiculous military prodigy he is to us. They also likely prioritize herding and grazing over farming, so their communities may look different--and they're a lot more likely to need preservatives or to have a more complex dietary regime (e.g. incorporating dairy, eggs, complete-protein pairs, or other similar things into most dishes).
This gives the idea of a culture where pride and legacy are dominant, where leaving your mark on the world is incredibly important--"going off to seek one's fortune" is extremely common in their society, not something only callow youths with no family ties pursue. It's also one where building to last matters (since, y'know, a single big argument could lead to burning a place down if you aren't careful...) and where "go big or stay home" is likely deeply-ingrained. In other words, it's something like a hybrid of "fairy-tale farmer's sons" and "the idealized Romans of legend."
