At KRyan's prodding, converting my comments to an answer:
The Rebirth was the period following the Blue Age, at the dawn of the Green Age, when the ancient halflings used the magic of the Pristine Tower (possibly in combination with some life shaping technique) to change most of their race into many new races to populate the new land created after the oceans retreated.
A complete list of Rebirth races has never been enumerated. The vast majority of sentient humanoids (and at least some extinct non-humanoids, like wemics), were spawned in the Rebirth, or the product of the blending (via mating or magic) of Rebirth races.
As KRyan notes, a notable exception to this rule are the gith, a corrupted remnant of an ancient githyanki invasion from the Astral, and don't count as a Rebirth race.
The following is a summary of what can be gleaned from "A History of Athas" in the Dark Sun 2E Revised Campaign Setting:
The best information we have available is from the listing of the Champions of Rajaat during the Cleansing Wars. The Cleansing Wars were specifically about eliminating the Rebirth races (aside from humans, initially, but Rajaat planned to eliminate the humans at a later point, after using them to wage the Cleansing War), so logically, any race with a champion assigned to them was a product of the Rebirth. There are 16 champions listed, one of whom replaced another, and had the same target race, for a total of 15 confirmed races spawned by the Rebirth; in some cases the status of that champion's war is given, in others it's omitted, but from monster manual and other sources we can nail down whether or not specific species survived the wars:
- Kobolds (dead)
- Ogres (dead)
- Giants (alive)
- Trolls (dead)
- Orcs (dead)
- Gnomes (dead)
- Pterrans (alive)
- Elves (alive)
- Wemics (dead)
- Lizardmen (thought dead, secretly alive at the Last Sea under the protection of the Mind Lords)
- Aarakocra (alive)
- Pixies (dead)
- Dwarves (alive)
- Goblins (dead)
- Tari (alive)
This list has been reproduced on the Dark Sun Wikia.
It is also stated that Rajaat himself was a Pyreen, which were also products of the Rebirth; no Cleansing War was launched against them, possibly because their numbers were so few they weren't worth a full champion, possibly because they were too powerful for anyone but Rajaat to fight.
Regardless, that gives us 17 confirmed Rebirth races (15 targets of cleansing, plus humans and pyreen), eight were "cleansed", one is thought to be extinct, but survived outside the Tyr region, and eight (five of whom are playable PC races) are widely known to have survived.
On top of that, it seems likely that half-elves and muls would qualify as Rebirth races; they're the products of natural (not magically assisted/altered) reproduction between two Rebirth races, and the feat refers to a heritage linked to the original halflings, which they would clearly share with their parents.
Half-giants are tricky: They're explicitly stated to have been the product of sorcerer king experimentation dating to just after the end of the Cleansing Wars, when the sorcerer kings were establishing/fortifying their city states, setting up the status quo that prevailed for thousands of years. They were created from humans and giants, but through magic, not mating; they can't breed with either parent race, only with each other. Arguably, whatever mad
sciencemagic went into their production would separate them from the lineage that traces back to the ancient halflings.
The other notable exception to the "basically everything comes from the Rebirth" are the insectoid races (thri-kreen, tohr-kreen, and their various subspecies). Though their exact origin has never been established, at least a few sources refer to them existing as far back as the Blue Age, just in tiny numbers (since they aren't well suited to living on an ocean world). The one thing that's clear is that they were not a product of the Rebirth; they benefited from the end of the Blue Age, and later, the end of the Green Age as the climate suited them more and more, but they evolved (or whatever D&D equivalent of evolution applies) from a species unrelated to the ancient halflings. I want to say (but can't find a source) that there is a reference to them possibly being the result of halfling experiments with nature; not created out of the whole cloth, but possibly uplifted from smaller, non-sentient insects to the form they have today.
Lastly, while modern halflings aren't a "product" of the Rebirth (just leftover original stock), they'd logically have as much ties to the ancient halflings, and should qualify for any feat based on ancient halfling heritage. There is an implication that modern halflings aren't quite the same as their Blue Age forebears (e.g. psionics were not common in the Blue Age, but are ubiquitous in modern halflings), so it's possible they're as much a Rebirth race as all the others, just more physically similar to the original race.
So on top of the races targeted for cleansing, we should add the following living races as products of the Rebirth:
- Half-giants (maybe, up to the DM)
So, short version: Every playable PC race aside from thri-kreen and (possibly) half-giants qualifies for the feat.