Who was the last Elf born in the Middle-earth?

The short answer is that the last elves born in Middle-earth were probably from a group that we never heard anything about, but there is no way to know for sure.

Questions about this, about the long-term fate of Middle-earth, pretty much cannot have canonical answers, because Tolkien did not publish anything about goings-on beyond the beginning of the Fourth Age of the Sun, and his views about the entire cosmology changed fundamentally over time.

At an early stage, Tolkien thought of his fantasy settings of Beleriand and Middle-earth (a name partially calqued from Midgard) as an alternate mythological history of the real world, possibly as as retold to Ælfwine, an Anglo-Saxon who had found his way to the the isle of Tol Eressëa, at the edge of Elvenhome. The ages would continue up to the present, and just as the First Age ended with the overthrow of Morgoth, and the Second and Third Ages terminated with great defeats of the second dark lord Sauron, there would be further dark lords and further wars. Tolkien suggested that the Second World War, with the defeat of Nazi Germany and the dawn of the nuclear age, might mark the boundary between the Sixth Age of the Sun and the Seventh.

As late as the 1960s, Tolkien was still thinking about having The Silmarillion as a fictional primordial history of the Earth. However, he realized that the early stages of the story would have to be completely revised, to account for the well-established facts that the Earth was very old, and it has been orbiting around the sun for its entire existence. Tolkien made stabs at reworking things this way, but he apparently decided that, since the story of the lighting of the world (first by the lamps, then by the trees, from which were created the stars, and finally the sun and moon) was such a fundamental part of the mythology he had created, he could not really do away with it without discarding most of the creative edifice he can created. So he decided that Middle-earth was completely fictional world—not the past of our universe, but still similar in some ways, so that the lessons of Middle-earth might be applicable in our world.

In the original conception, Tolkien wanted the cosmology he created to be consonant with other European folklore, especially northern European folklore. Elvish beings (potentially both light and dark) are an important element in many European myths, and Tolkien made them a central focus of his stories. By Tolkien's time, the era of the elves was over—and had been over for a very long time. Yet there were still old tales of fairy circles, of people being put to sleep, of the Wild Hunt—leftovers of the mythic age that had persisted for a long time, even through the ages in which mankind had dominated the world.

In Tolkien's stories, most of the elves had departed, via the Straight Road, to the Undying Lands, but there were always the Avari, or the Unwilling—those who had never responded to the summons of the Valar and embarked upon the Great Journey. By the heyday of mankind, those were the only elves left for humans to meet, and over time, we met them less and less. Tolkien explained this as the elves gradually fading away; although they were immortal, they lacked the vibrancy to exist forever in Middle-earth. The elves of human folktales were the last, not quite faded Avari, and as the very last of the Avari finally did fade, their influence on the world came to an end.

However, Tolkien wrote practically nothing about the Avari. (I believe he mentions the name of one of the Unwilling, and that is it.) We know essentially nothing about them after they were left behind at Cuiviénen after the War for Sake of the Elves. Presumably, the Avari continued to live and reproduce for a long time before the last of them faded away, but we no nothing about this quantitatively. I think the last elves born in Middle-earth were thus Avari, but it is impossible to say when those births happened. At some point, the last lingering elves would lose the potency to procreate, as part of their fading, but there is no information about when that actually occurred.

Moreover, after Tolkien gave up the conceit that Middle-earth was the mythic past of our world, things became even trickier. As I said, I think the last elves born in Middle-earth were Avari, who were still there after the second departure of the Eldar. However, after Tolkien changed his mind about the cosmology, there was no longer any fundamental reason to insist that the Avari had to have remained a (somewhat) thriving community into the Fourth Age and beyond. Because they did not need to stick around to become the fae that bedeviled premodern man, the Avari could actually have been wiped out entirely—perhaps even in the Elder Days, before the War of Wrath and the overthrow of Melkor. In that case, the last born elves in Middle earth would clearly not be Avari, but we do not know who there were instead.