Short answer: this was an accident.
Tolkien was asked this question, and he answered it in Letter 257:
From The Hobbit are also derived the matter of the Dwarves, Durin their prime ancestor, and Moria; and Elrond. The passage in Ch. iii relating him to the Half-elven of the mythology was a fortunate accident, due to the difficulty of constantly inventing good names for new characters. I gave him the name Elrond casually, but as this came from the mythology (Elros and Elrond the two sons of Eärendel) I made him half-elven. Only in The Lord was he identified with the son of Eärendel, and so the great-grandson of Lúthien and Beren, a great power and a Ringholder.
The Letters of J.R.R. Tolkien 257: To Christopher Bretherton. July 1964
A character with the name "Elrond" dates back to the earliest versions of The Silmarillion, but he has more in common with later-introduced character of Elros. A draft dated by Christopher Tolkien to 1925-6 reads:
Elwing cast the Nauglaf-ring into the sea and leapt after it, but was changed into a white sea-bird by Ylmir1, and flew to seek Eärendel, seeking about all the shores of the world.
Their son (Elrond) who is half-mortal and half-elfin, a child, was saved however by Maidros. When later the Elves return to the West, bound by his mortal half he elects to stay on earth. Through him the blood of Hurin (his great-uncle) and of the Elves is yet among Men, and is seen in valour and in beauty and in poetry.
History of Middle-earth IV The Shaping of Middle-earth Chapter 2: "The Earliest 'Silmarillion'" Sketch of the mythology with especial reference to the 'Children of Hurin' Part 17
When we come to about 1936 or so, we encounter early versions of the Fall of Númenor; Elrond has a brief mention here, where he appears to still be an early version of Elros rather than the Elrond we know and love:
[Númenor] was built first by Elrond son of Eärendel, whom the Gods and Elves chose to be the lord of that land; for in him the blood of the houses of Hador and Beor was mingled, and with it some part of that of the Eldar and Valar, which he drew from Idril and from Lúthien. But Elrond and all his folk were mortal; for the Valar may not withdraw the gift of death, which cometh to Men from Ilúvatar.
History of Middle-earth V The Lost Road and Other Writings Part 1 "The Fall of Númenor and the Lost Road" Chapter 2: "The Lost Road" (iii) The second version of the Fall of Númenor §2
Elrond moves closer to our conception of him in a later text:
Yet not all the Eldalië were willing to forsake the Hither Lands where they had long suffered and long dwelt; and especially in the western isles of the Land of Leithien. And among these were Maglor, as hath been told; and with him for a while was Elrond Halfelven, who chose, as was granted to him, to be among the Elf-kindred.
History of Middle-earth V The Lost Road and Other Writings Part 2: "Valinor and Middle-earth before The Lord of the Rings" Chapter 6 "Quenta Silmarillion" The Conclusion of the Quenta Silmarillion §28
Christopher Tolkien doesn't conveniently date this manuscript, but I suspect it came long after The Hobbit was published in 1937. In fact I would suspect that the development of Elros was a direct result of Tolkien's error in The Hobbit; Elros was introduced to take the place that had previously been filled by Elrond, so that Elrond's appearance in The Hobbit could be reconciled with the rest of the mythology.
1 Early incarnation of the Vala Ulmo