Ahem. Hi. Hello. Hello.

It's the Librarian here, fresh from the Whoniverse. The Doctor picked up some bad Hunger Games characters on his Timey-Wimey detector (don't ask, it's wibbly-wobbly) so he shoved me into this universe to help sort them out. It hurt a bit, mind you. Felt a bit cramped, kinda like I was being pulled and stretched and squashed all at the same time. Then there was this weird jolt – and the Doctor kicked me out of the TARDIS.
But anyway! Here I am.
So I'm looking at your character, and tsk tsk. You need a name, fast. Sweetmeat Applepie Honey Melon Dewdrop won't cut it, even if she's from somewhere as out of it as District One. So let's start there, then.

District 1
Ah! District One! This is where all the luxury items are produced. It's the wealthiest of the districts, and prides itself on having many victors. Naturally, being full of money, the names of their children are going to reflect that. Of course, this means that oftentimes you'll find something along the lines of Toffee Chickenwing Buttersc- nope. Never mind.
Canon names: Augustus, Cashmere, Glimmer, Gloss, Marvel
Looking at these, we can see that the names from District One are going to be luxury items, rich-sounding adjectives, or Greek or Roman names. Think along the lines of precious gems or feminine products (just don't go for Tampax. Just imagine the jokes if your character gets an axe to the face!)
Overused names: Diamond, Shimmer, Silk
Examples you can use: Lace, Pearl, Garnet, Atalanta, Blaze, Rouge, Antoinette/Antoine, Platinum, Blade, Brilliance, Valour, Apollo, Beau, Iasus, Blanche, Sterling

District 2
District Two is the main producer of masonry for Panem. It produces its fair shares of Career tributes and victors, and its tributes tend to be far fiercer. Steer towards more Roman names for this one, or names that reflect building materials. Even go for the occasional natural name. Try and make their names similar to those of the Capitol (as they're the Capitol's lapdogs) but more like ones you'd hear for a fighter. If that makes sense. Which it doesn't. But whatever.
Canon names: Brutus, Cato, Clove, Cray, Darius, Enobaria, Lyme, Purnia, Romulus
I've yet to see many characters from District Two, despite it playing such a huge part in the rebellion. So don't worry about getting an overused name. Just don't go for Clover, because duh. You'll get your head smashed in with a rock for copying Clove.
Examples you can use: Nolan, Remus (I cri everytiem), Jason, Victor, Mason, Odius, Titus, Titan, Blade, Alexis, Lupin (ffffff), Dagger, Copper, Juniper, Artemis, Albus, Augustus, Caelia, Claudia, Riddick, Ply, Slate, Flint, Serena, Smith

District 3
You can thank District Three for all of your technology, from the television in your house to the camera that the Capitol is using to stal- I mean, to make sure thieves are caught. Here, go for futuristic names, but try and make it subtle. Don't call your character Samsung or Cellphone. That's just stupid.
Canon names: Beetee, Wiress
Examples you can use: Fleck, Flash, Tallie, Chip, Cable, Wyre, Eleria, Dot, Pixel, Aurora, Link, Bolt, Mac, Render, Dell, Binary, Console

District 4
No, that's not the smell of fish and chips. Welcome to District Four, the fishing district! This is the last district to train career tributes, and as such their people are often strong. These names can be fairly normal, so don't worry about being too flash. Water-themed names are good, and so are the names from water-based mythology. Try and steer away from gods and goddesses, though. You won't see any self-respecting person walking down the street who's called Poseidon. Unless he is, of course, Poseidon.
Canon names: Annie, Finnick, Mags (possibly short for Margaret or Magdalene),
Examples you can use: Ariel, Nemo, Una, Muriel, Adrian, Caspian, Coral (but please be careful of this, it's a little overused), Wade, River, Amelia, Cress, Brook, Pier, Hook, Cliff, Morgan/Morgyn/some variation thereupon, Sebastian, Lynn, Pearl, Cordelia, Tyne, Finn

District 5
Who powers your lights? Who powers your television? Who powers the screens that air the Games every year? That's right, District Five. Again, choose futuristic names, but don't go overboard.
Canon names: Finch (I suppose, but it's not really canon), Porter Millicent
Examples you can use: Fuse, Wyre, Spark, Atom, Newt, Watson, Nina, Electra, Tesla, Brighton, Tessa, Faraday, Lux

District 6
This is where you get your transport from. Cars, hovercrafts, trains… they're all made right here. There aren't many character names to help us along at all, so feel free to go from names relating to the district's main industry.
Canon names: Titus
Because of Titus, you might also like to choose Greek or Roman names, but I think it would be quite rare to see those in District Six.
Examples you can use: Aston, Cooper, Ford, Martin, Apollo, Carter, Cara, Harrison, Dakota (this one would be good because of the whole America thing, and can be used for other districts), Axel, Verona, Mercedes

District 7
Aha, good old District Seven. Paper, wood, sawdust. With this, tree names work, but try and make them interesting. And just by looking at Johanna, we can see that not every District 7 female is named Willow. They're much like District Four in that they have relatively normal names.
Canon names: Blight, Johanna
Overused names: This is where we start seeing more and more overused names. Willow is really popular, and so is Oak.
Examples you can use: Aspen, Leif, Ivy, Rowan, Mable, Shred, Ewan, Monty (see how many people get the reference!), Hunter, Finch, Ren, Linden

District 8
Here you get all of your cloth and thread and other textiles. Because of this, you can go for slightly more frilly names. And if Lahcé is a bit ridiculous, then go for natural-sounding names.
Canon names: Bonnie, Cecelia, Paylor, Twill, Woof
My own character names: Spindle
Examples you can use: Thread (be careful with this one, we all know what Thread's like), Penny, Spindle, Lacey, Weaver, Angora, Silk, Bunt, Sheer, Thimble, Bobbin, Bernina

District 9
Grain. Wheat, rice, barley, you name it. You get that from District Nine. There are no canon names here, not even from Capitol Couture, so go for names that relate to wheat or havesting.
Examples you can use: Maisie, Barley, Rye, Silo, Bran, Juno, Farro, Emmer, Amaranth, Demeter, Cybele, Barric, Granger

District 10
Ten is my favourite number, but probably not my favourite district. Here you get your livestock. Cows, sheep, pigs, goats, it all comes from Ten. Even with only one canon name, we can see that the names from District Ten don't necessarily relate to the industry.
Canon names: Dalton
Examples you can use: Ari, Philip, Fauna, Bram, Cat, Pelt, Nolan, Briar, Cass, Sable, Dyana, Talon, Auberon

District 11
And thus we step into the very dangerous territory of overused names. That's right. We're in District Eleven. You're very likely to hear natural names here (Rue is a flower and Thresh means 'to separate grain from a plant') so it's a safe bet to stick to those. Of course, don't go for names that crop up often (aha, great pun).
Canon names: Chaff, Martin, Rue, Seeder, Thresh
My own character names: Hazel (yes it's overused, I know. I was twelve. Shut up.)
Overused names: Willow, Rose, Hazel
Examples you can use: Bramley, Lark, Finch, Twill, Loren, Barley, Ambrosia, Pyrus, Fescues, Olive, Aster, Lennox

District 12
Aaaaaaaand finally, we come to District Twelve, the poorest of all the districts and the supplier of coal.
Most tributes you'll see in fanfiction seem to come from here, probably because it's the district we hear the most about, and of course, everyone wants to be like Katniss. Except maybe the whole Prim-go-boom-boom part. But I digress.
If you look at the canon names, most of the characters don't have natural names. So, go for those if you want, but otherwise pick something that sounds like your character's personality. Here we realise what a hypocrite I am. Please remember that I was twelve when I created these characters, so there are definitely overused names in my story.
Canon names: Bristel, Delly, Gale, Greasy Sae, Hazelle, Haymitch, Katniss, Leevy, Madge, Maysilee, Peeta, Posy, Primrose, Ripper, Rooba, Rory, Thom, Vick
My own character names: Bailey, Burnet, Ember, Kaitlynn, Melia, Posy, Willow
Overused names: Cole, Rose, Willow, basically any flowery name.
Examples you can use: Burnet, Myner, Slater, Jewett, Ash, Colton, Arden, Blair, Colman, Hilt, Dusty, Mist, Canary (only use this for really poncy merchants' kids)

How to Come up With Your Own Names
Use odd spellings. Caitlin = Kaitlynn, Burnett = Burnet, etc. Is become ys, ls become lls, cs become ks.
Google/use another search engine to find (just don't use Explorer. If so, you're on your own) synonyms for words. So if you were to use Glimmer for a District One girl, since that's already taken you could use Gleam, Shine, or Glint. Fierce could become Savage or Grim. That kind of thing.
Combine words. That's how I came up with Melia. I was lying in bed one night back in 2014, and being the cheerful little twelve year old I was, I was thinking about malaria (don't ask). Somehow from that I came up with Melia (pronounced Me-lye-ah, not Amelia without the A). And it seemed perfect for the character, so I stuck with that. I've now become so attached to the name Melia that I can't imagine her being called anything else.
For District One, look around your bedroom. Perhaps your underwear brand would work? Just imagine: Klein, the Victor of the 69th Hunger Games, won by strangling his fellow tributes with his boxers. Okay. Don't imagine that. But think of toiletries. Makeup brands. Book titles, for Gale's sake.

DON'T EVER USE NAMES YOU'D SEE TODAY! The only exceptions to this are Districts Four and Seven, and even then you're pushing it. There've been too many fanfics where I've seen "As always, ladies first! Chelsea Wilson! And now, for the boys! Justin Andrews!" NO. First, surnames tend to be strange. You're not going to see any Sparkle Glitter Buttina Wilsons walking around in District One. Unless her parents hate her. Sparkle Glitter Buttina Sattine might be more plausible.
Melia's surname is Cressview, which sounds a lot like the ones you hear in District Twelve. That was the point. And her friend Bailey is Bailey Birling. They both sound like regular people from Twelve. So I give you permission to use both of those in your fanfiction.

So there we have it! Now you know how to come up with names for your tributes. If you'd like, send me your character names and I'll see if they work, or I can help you come up with even more names.
Next up, we'll have appearances. As Effie would say, "Chins up, smiles on!"
Don't forget to bootysmash that favourite button, add to your follows so you can catch updates like Prim caught fire, and review like a Gamemaker!

Happy Hunger Games, and may the odds be ever in your favour!