OD&D is pretty clear that "there is no reason that players cannot be allowed to play as virtually anything, provided they begin relatively weak and work up to the top..." (Men & Magic, p. 6).
Thus, I'll probably work up other races to add into the basics as I go--I've always been a fan of canine races, for example. But for now, here are the niches into which the core races fit. It should be noted that these are campaign notes, not game ones. In other words, Humans, Elves, Dwarves, and Hobbits have the same racial abilities (mechanically speaking) as listed in Men & Magic.
Humans: The most populous single species in the world, humans are currently battling to "tame" the wilds of the Wasted Lands and create a new civilization from the ashes that came before. The godswar, so many thousands of years ago, devastated the lands and drove more than a few sentient species to extinction. But humankind, ever tenacious, has once again crawled from the ashes of cataclysm to reclaim the world, battling against what sometimes seems a nihilistic slide into oblivion. Humans span the range of emotion and moral proclivity, from lighthearted, optimistic, and reverent to dark, violent and power-mad. Humans were created by the Ancient Ones as all-purpose workers, soldiers, and playthings, so they are adaptable and determined in ways that other races are not.
Dwarves: dour and brooding, dwarves prefer darkness and barrenness to life and growing things. This doesn't necessarily make them evil, or imply that they don't have a respect or reverence for life; it just means they are more akin to the rocks they mine than to other life forms. Dwarves were created by the Ancient Ones to be miners and masons. So they did not suffer when droves of their kind suffered in mining collapses and building incidents, dwarves were created without strong emotional attachments. They tend to be driven by two extremes: anger and logic. Dwarves are experts at putting pieces together, analyzing situations, and determining the most logical path to follow, but their rage often gets the better of them, particularly where corrupt subterranian species such as goblins, orcs and kobolds are concerned. After being betrayed by the elves during the Godswar, the dwarves went to war against the elves, and their battles forever scarred the land, forming a war within the greater war.
Elves: Elves are a dark species, related in many ways to goblins. They are possessed of a haunting, etherial beauty and seductive nature. They are feared and renowned as spies and assassins. It is said that in the early days, humans attempted an uprising against the Ancient Ones, to shed the chains of their slavery, and the Ancient Ones desired to ferret out and destroy the leaders of the insurrection rather than wiping out all human beings. So they turned to the creation of a race that could infiltrate and seduce humankind. Their first efforts went poorly, resulting in goblins and orcs. Finally, they succeeded in their efforts, and the elfin race was born. Elves live in the forests and are generally nocturnal, preferring the shadows of night and silver moonlight to the bright colors and yellow sun of day. Still, operating in daylight causes them no more than minor discomfort, and a feeling of being out of place. Despite being expert thieves, spies and assassins, elves have a strong racial sense of honor and duty to those they consider family, making them strong, if always questionable, allies. During the Godswar the elves were set to play all sides by various masters and contractors, and perpetrated a great betrayal of the dwarves. The exact nature of this betrayal has been lost to time and legend, but it resulted in a mighty war between the elf and dwarf races. Tension exists to this day and the elves, though having softened and gained a new respect for life and nature through observation of hobbits, are still sly and crafty and in control of many spies', thieves', and assassins' guilds.
Hobbits: Hobbits, it is believed, were a first attempt at creating humans. Unfortunately, their creators went a bit overboard when making them docile, and the result was a carefree race possessed of a great love of food, drink, and smoke. Still, hobbits came about instilled with an abiding love for all things that grow, making them useful to the Ancient Ones as cultivators of land to sustain the other races. Eventually, the elves were set to watch over the hobbits and ensure that no harm came to this diminuitive race. For a long time the hobbits were completely unaware of their elfin overseers, but eventually, certain elves came to be fascinated by the little people and made contact. It is said that the relationship between hobbits and elves is all that kept the elfin race from falling into true and complete darkness. It is because of the hobbits that the elves gained a connection with the forests and a respect for life.
So there we have that. I thought to take a slightly different approach to elves and dwarves, making dwarves a more paradoxical race that tries to temper rage with stone cold logic, and elves a darker species with a pension for professional killing and infiltration. If I were to make any alterations to the core species traits in OD&D based on this, I'd probably give the Elves +2 to Charisma scores and/or a bonus on reaction rolls to reflect their mystery, beauty, and seductive nature.