I know this lineup is impossible in a unity edition, but I'd want:
Changeling: Reflavor them as part of the faerie tradition, truly crib-swapped individuals who realize their fey heritage and are either driven out and persecuted, or else stricken with an irresistible wanderlust, or need to return to their magical homeland. I'd route for some iteration of the Feywild to be in 5e, and having them as some sort of living key or one of the easier means of entering and exiting the place. There's just so much mythology and fantasy wrapped up in Faerie, I think this is a good place to vet it while still allowing a player to remain, at least for early levels, somewhat ignorant of how the DM is going to handle the Feywild and faeries. The Changeling is part of that world, but not intimately from the get go. I'd also want Changelings to have the opportunity to grow into full nobility, Sidhe or something along those lines.
Dwarf: Dwarves are so great no matter the setting, rugged, stout people that somehow manage all these shared fantasy connotations without pulling you out of any particular setting. They just fit. And better yet when you start fiddling with them a little to make them more campaign specific. Just a great race out of the box or after some odd treatment or another.
Eladrin: I'd prefer the 5e PHB Elf to actually be an eladrin, and bring the mystery, aloofness, charm, and magic back to a race who's, in my opinion, gotten too close to humans in the last two editions. I'd be fine with agelessness, the ghostly quality of living in sheltered forest cities, and their penchant for walking unseen paths (teleportation). Let settings dictate the eladrin as a more rugged 'elf ranger' type. Ironically, I think I'd want to remove the fey connotations from the eladrin, though, and keep them as evidence of the beauty, majesty, and magic of the prime material worlds.
Gnoll: Monstrous races should be represented right at the get go, and of all the monstrous races I feel the gnoll has the most potential in the wilds of any particular setting, and is, in my opinion, more flavorful and palatable than the orc. They could really be positioned into that space the 'elf ranger' used to occupy, a more grounded, wilderness race that conflicts with civilized worlds and more supernatural ones both, and reflects worldly nature in their bestial natures. They could bring in a lot of shamanism, totemism, and animism, and provide a great way to involve Primal Spirits into character's immediate lives (which I do hope make a 5e comeback).
Gnome: I want a flavorful and interesting small race, but I always thought halflings and gnomes sort of fed off of the same design space in many ways which made them both weaker for it. Smallness alone gives a race a sort of charm, but there should only be one initially. I vote gnome. I know there's some assumed setting in what follows, but make gnomes the wily pranksters and elusive thieves of the borderlands, not quite belonging in the wilds with gnolls, or in the distinguished company of eladrin, nor welcomed in the civilized society of man and dwarf as anything other than second-class citizens. They should be the race that lives between and underfoot all the other races, in barn lofts, under cellars, in small thieving troupes, or as drifters. I don't know if they should be fey, but if so, something like household fey, inherently low magic save a few keen tricks and a gift for not being caught. Maybe I really want a Brownie or Boggart or something here, but definitely not straight up Kender.
Goblin: Too good to have gone five additions without getting the royal PHB treatment. So many people would play these guys it isn't even funny. They, like dwarves, possess this sort of ingrained awesome that comes from countless treatments over fantasy mediums since myth. Maybe they're idiots, or cunning night-stalkers, or of the teeming hordes which threaten to overwhelm civilization (I do love all the different Magic the Gathering treatments over the blocks). They can be spun so many ways, and all of them are somehow compelling.
Half-Ogre: Products of ogres and humans, for a number of reasons, resonate with me far more than of orcs. That design space has been tread, whereas half-ogres have not (at least recently, or on the onset). I mean they really hit the point home of union between bestial giant and human. They're larger than half-orcs, stronger, just as savage and conflicted, and still have all the roleplaying wealth. They also allow for more re purposing as any sort of half-giant or maybe pure-blood giants and ogres. Lets make them Large, too.
Human: I am one, and I have a soft spot for them. They are usually a major focus of nearly every setting. They are a staple I don't think anyone wants to be rid of, as versatile our actual history shows us. They can be anything, different races unto themselves, honestly.
Shifter: Despite the gnoll in my lineup, I believe the Shifter (under a new name, please) has a spot in the 5e PHB as one of those 'something more than human' types, namely a lycanthrope. It's a strong fantasy archetype and a desirable one. I'd give them the power to change anthro or the actual animal right off the bat, too. Why hold off on that? I'd rather we give players such racial tools from the onset, and their skill and player ability to use well progresses from there. I honestly don't see any sort of imbalance in becoming an ordinary animal or a more physically adept humanoid. Of course it'll scale, but yes, put the lycanthropes in the PHB. They could also fill some of that gritty player space afforded a low magic quality I so enjoy.
Vampire: They're here to stay. They're ingrained into the culture, and into rpgs, and even into D&D. Thank Ravenloft, I suppose. Thank every vampire character, setting, book, etc. I rather like them myself, though minus a few of the more gimmicky weaknesses (garlic) but fine with others (aversion, not destruction, to direct sunlight and maybe difficulties with opposing alignment holy symbols). I also like the idea of an undead race in the starting lineup, one that has to be at least a little dark in its eating habits. They also cover a huge design range, from vast, dark, aristocratic empires, singular baronies and fiefs, to lone stalkers as part of an unseen culture spanning the entire setting. They could also pull the Shadowfell's equivalent into greater focus, which I like.
Wilden: Another name I'd change, but a concept I'd definitely keep in some capacity. A plant race is something I'd like to see done with respect and a creative touch, that will also be supported throughout the edition and not a one-and-done. These types of races deserve the respect all the others got at some point. Maybe it comes in the form of the recent Heroes of the Feywild hamadryad, or Guild Wars II's Sylvari, but I want something not animal involved here.
On the fence:
Tiefling: As someone mentioned, fiendish ancestry is very bad ass and sings of Sword & Sorcery. It has a place in a lot of settings and campaigns, and is a desirable choice from the outset. I wouldn't mind expanding the bloodline beyond infernal, and maybe keeping the uncertainty of the exact affiliation as part of their mystery, at least for a time. You could have tieflings factor heavily into fiendish politics or Blood Warring, or into the legacy of Bael Turath style fallen or corrupted empires. How monstrous they look should be up to DMs, though I wouldn't mind a progression that starts very subtle and becomes increasingly heinous.
Deva: While being the offspring of an angel is cool, and adding Nephilim and biblical aspects to this race would also be cool, I also dig the reincarnation aspect that comes with the 4e Deva. I think those cosmological specifics should vary from setting to setting, but some form of promised one, or enlightened one, again another one of those 'more than mortal' types skewed toward heavens and higher planes is a compelling addition. Like the tiefling, I wouldn't mind the lineage of any Deva or Nephilim to begin very subtle and become more and more pronounced the higher in level they go.