My own answer to this is... oh, boy, long.
Gnolls: 4e's Playing Gnolls article in Dragon Magazine #367 took a race that'd actually been a PC option since Basic and finally gave it some fluff that amounted to more than "hairy orc slavers".
Dragonborn, 4e: I found dragonborn filled a long-missing niche in being a corebook dragonman race. I mean, seriously, the only dragonman race we had before 4e that wasn't the half-dragon was Dark Sun's Dray, and who remembers them? Add to it that I loved the fluff about Arkhosia, and they were a shoe-in. It's a crime what 5e's done to them.
Tiefling: Pre or post 4e, either works for me.
Lupins: I didn't know these guys existed until I found out about them in Dragon #325, but the picture it gave me of swashbuckling wolf-people is too fun to let go of.
Aranea: Why these guys haven't been made a proper PC option since Red Steel, I don't know. Sapient giant spiders who can shapeshift into partially or wholely humanoid form, naturally adept in magic, but who completely avert the stereotype by not being a bunch of evil man-eating monsters.
Kobolds: Just so much fun when you play with kobolds. Goblins too, but kobolds are generally more fun to play with.
Sphinx: I know it hasn't been a valid race since Basic, but even so, it's perfect for those more out-there fantasy settings. Hell, there was a series of Greyhawk novels where a sphinx was part of the adventuring party.
Kitsune/Tanuki: Really seem like they'd be fun for an Oriental Fantasy Adventures campaign... shame that the Hengeyokai race in D&D has always been such a one-trick pony.
Phanaton: It's a freaking raccoon-monkey-flying squirrel mix 'n' match critter. Perfect for gonzo/weird fantasy worlds.
Deva: Angels who gave up the heavens because mortal life is so much richer. You seriously telling me that's weaker than "my granddaddy was an angel"?
Shadar-Kai: Because they're a unique, awesome humanoid salvaged from what was a pretty generic asshole emo-fairy monster.