Full disclosure: I've only read the first book, and that was quite a while ago, and I didn't much care for it.The following is largely based on stuff you guys have said here, a brief look through the series Wiki for terminology, and an admittedly piecemeal classical education. If it's incompatible with HG canon, feel free to totally ignore it.
---
These Tribute chaps, chapesses, and nonbinary people of general chapulicity, are basically gladiators. They've even got a vaguely-defined laurel crown being dangled in front of them like a carrot before a donkey. Indeed, from what I've been able to glean, there's a whole Fall of the Roman Empire thing going on with the series, from the bread theming being an incredibly laboured reference to the concept of "bread and circuses" to the Capital (i.e. Rome) being this kind of decadent hellscape of flamboyance and sybaritic menace... which might say more than Collins is intending to about the aesthetics of camp, but that's for another day. You have, every year, one massive balls-to-the-wall spectacle of ultraviolence wherein a bunch of provincial types beat the tar out of each other in a changing environment. I'd like to remind people that the floor of the Coliseum was occasionally flooded so that gladiators could engage in naval battles.
So far, so Roman. However, there's a smattering of something else in the mix, and that something smells of professional wrestling with considerably less Health and Safety input. It feels artificial and stereotyped to the nines, which I can only assume is the point. So you've got the potential for narrative enhancement in with your gladiators, and if there's one thing people love, it's a heartwarming underdog story.
So you rig it.
See, the first rule of writing believable villains is that they shouldn't think they're the villain. Who does? Therefore, we can assume that the citizens of the Capitol don't believe themselves decadent petit-bourgeois representations-of-LGBT-stereotypes-to-be-got-rid-of-by-the-backwoods-small-town-yokels-of-the-purported-"Real-America" - er, what were we talking about? Villains. Right. The people of the Capitol are just that. People. Not intrinsically evil or anything like that. Just... people trying to have a little entertainment in their lives. So you have the perfect setup for a "heartwarming" tale of triumph in the face of adversity, so you pick a tribute with an obvious physical disability and you give them... perhaps a little more help than they need.
I admit, I'm basing this off the whole Pegleg Peeta discussion that happened elsewhere, but it seems like someone with a missing leg or whatever would be exactly the kind of Tribute people would get behind in a narrative about love and honour and eventual, inevitable tragedy. I don't use the word lightly, either - you can put together a narrative straight out of Greek tragedy by giving our hypothetical Hopalong Cassidy here a bit of extra leeway. Maybe he knows a particular type of terrain from his home district - so you base a few of the earlier rounds off that. Maybe he's a good shot and a patient hunter - so you put in scenarios where fast-moving combat is discouraged. You create this feeling of tension, focusing on our Tribute, ratcheting it up - will it be this week he dies, will it be this time, oh God I can't bear to watch it pass the popcorn Darren. You pick the other Tribute to have the final battle, the one who's also good at this, and you save them for last, and you have the final battle wherein the one-legged Tribute has a definite but not obvious disadvantage to their able-bodied counterpart.
And when Jake The Peg here loses, what happens next?
Well, that's how I'd do it, anyway. Feel free to rip it apart. =]