Dwarves -
So many things to do with dwarves. Hell, I would love to see dwarves actually be the unicorns of a setting for once; very rarely seen, revered and feared at the same time, and highly valued. But since that's never going to happen...
In the lore for the Shadowbane online game, Dwarves are a static race, fashioned from stone by their creator-god at the dawn of time to tend the earth's forges. They are never born, and they never die. All appear male, but none of them have any gender in any meaningful sense.
This was a very cool 'spin' on Dwarves, explained the dearth of female dwarves in fantasy literature, and also made them rare and precious, each dwarf being a unique entity, formed when the earth was new.
It would be easy enough to tweak this to work mechanically (and not be unbalancing, with 'newbie' dwarves having millenia of experience). Simply say that the race has spent their entire lives underground, among other dwarves, their immortality and ages of wisdom are ultimately meaningless, mechanically, as they existed for millenia in a changeless state (effectively unable to learn or gain EXP, performing rote tasks). Some strange event has occured in the depths of the earth that has prompted some dwarves to throw down their picks and journey to the world above, 'awakening' from their stony slumber and joining the other races of man.
What drove them from the depths? Are they hunting for something up here in the surface world? Is there some secret agenda at work? Do the dwarves even know, on a conscious level, why they have abandoned their sacred charge to walk among the surface races?
Or have they finished whatever task they were fashioned to perform (or perhaps just the first stage of it), and are now 'at liberty' (perhaps only 'for a short time, a few centuries' before they have to return to the underforges)?
Every 'core' race could do with some tweaking;
For one game, I made Gnomes into gypsies, travelling the world in brightly painted wagons, famous entertainers, tricksters, soothsayers and animal trainers. They claimed to have a kingdom somewhere, but would never share the secrets of it's location. Their arcanists claimed to be tutored in the 'Invisible College of Thaumaturgy,' which *did* exist (and was kinda/sorta invisible, being an extradimensional space like a Mordenkainen's Magnificent Mansion, complete with illusory tutors and spellbooks, all concealed within one of their brightly painted wagons).
In a 2E game, inspired by a Dragon article about 'people under the hill,' I made elves into fey creatures, immortal, jaded, dissipated, and prone to terrible rages and heart-wrenching displays of grief, as their passions burned hotter than fire. An elfs life among humans would be like the life of Elric, forever mourning the friends that he has unintentionally slain. Creatures of chaos made flesh, the elven people live their immortal lives for the present. Every whim is acted upon, every regret is forgotten by the new day, when the elf may make the same spectacular mistakes he made the previous day, all over again...
Halflings? I dunno. I picture them less as Kender-nomads and more as Hobbit-stay-at-homes. They would have the most civilized communities, with common sense 'laws' enforced by good-natured sherriffs. Their communities would blend into the surroundings, simply because of the sense of security they get from living in underground homes, and well-tended fields would surround their townships. Being sensible folk, and surprisingly torn between insular and gregarious, when a town got 'too big,' whole families would pack up and establish new communities. There was no set law or procedure, and, quite often, this would just 'happen' when one family got fed up with their neighbors, as the stress and strain of a community growing 'too close for comfort' begins to lead to fights and arguments and other stresses off too many people packed into too tight an area. As a result, new halfling communities would almost invariably be settled by two sorts of families. Troublemakers, who left because they'd burned their bridges, or curmudgeons who claimed that the 'community was going to hell in a handbasket anyway, so we got out of there!' All throughout the lands would be quaint little Halfling farming villages, selling their wares to each other and the 'Big Folk,' living their lives. Dwarves, Elves and Humans might be surprised to know the true population of the Halflings, as they only see a little village here and a little village there, not realizing how quickly those 'little villages' add up, and that the 'little guys' outnumber some of their neighbors, and have their nations not only surrounded, but thoroughly infiltrated, and more dependent than they realize on Halfling-made bread and beer...
More unusual races can go even further afield from typical depictions.
The Warforged could be replaced with the Forgeborn, dwarven-created constructs that have furnaces in their bellies, and need to 'eat' wood and oil to keep their fires burning.
Changelings can be replaced with Skinchangers, who have the ability to slither out of their skin and look like a different person. At 1st level, they might take 10 minutes to do such a thing. By 5 HD, they can slither out of their skin as a full-round action (but it still costs them somehow, until they take a certain amount of time to devour the shed skin). Their 'true form,' in which they are never seen, is serpentine, and even when 'wearing another skin,' they have keen vision and the ability to 'squeeze' as a creature one size class smaller than themselves, due to their flexible form.