Halflings
Thanks to "The Hobbit" and "Lord of the Rings," halflings are traditionally expected to be rogue-class characters such as thieves, rangers and, occasionally, dual-class fighter-thieves. Halflings are actually quite proficient with these character classes and there's nothing wrong with playing them this way, but it might be more interesting to turn on its head the idea of the sneaky halfling. Consider playing as a wizard with a specialty, like a necromancer; perhaps an evil warlord killed all this halfling's countrymen, so he set upon a quest to become the world's greatest necromancer to raise his people from the dead and exact revenge upon the warlord with his undead army.
Elves
Long-lived, stoic guardians of the forest, elves are almost always portrayed peaceful, logical stewards of nature. But what if one were abandoned in the woods and adopted by a tribe of wandering barbarians? What kind of adult would he become if raised in a culture of blood lust and berserker rage? How would his superior strength, stamina and eyesight benefit his adoptive tribe?
Humans
Humans tend to be the jack-of-all-trades in any D&D campaign. They are the only race that can play as any character class, so, as with real life, it's easier to find humans performing any job imaginable and harder to create an iconoclast based on an odd choice of class alone -- this is where good back-story is most helpful. Perhaps a human character could see himself as the avatar of a god and act accordingly toward others, or have a quirky traveling companion he met under extremely odd circumstances. Work with the person running your game to see if you can agree upon some outstanding physical or mental attributes to set your character apart from the rest of humanity.
Dwarves
Dwarves east, sleep and breathe underground, so it's relatively easy to create a unique dwarven character -- just give him a crippling phobia of the dark or enclosed spaces to get him out of the mines. Don't let him become a one-dimensional character, however; flesh out his back-story with a life on the surface world. Perhaps he fell in with a high-brow crowd and became a musician. Dwarven bards are a rare sight, indeed.