Players could pick love, age (12-18 yrs), race (Caucasian, African-American, Hispanic, Asian, etc.), District, facial features (scars, blemishes, etc.), hair style and color, face structure (with a system similar to Oblivion perhaps), body type.
Weapon specialties, anyone?
A cutscene could introduce the player to the Capitol, where tribute training could act as a brief tutorial.
then, you're free to join matches in various maps and such. Players could be grouped with other players in similar structure to a lore-friendly Games, with a male and female from each district.
Last player standing wins in-game money which can be spent on HATS