Couple of thoughts....
Champions for Honor/Odium could be Humans/Listeners respectively. Either as a whole or individually. But that seems too simple/obvious.
Parallels between the Heralds "Right of Challenge" and the contract that Odium is bound to. Characters will quarrel over it because it's not actually taking away his power(a la Sadaes), but it's a rule that he's bound to and must accept the precepts. This is probably what broke the Heralds in the first place...this never ending fight that doesn't actually mean anything because it's a game. I imagine at the end of the fight a boon will be granted. It could be that the original boon was to seal away Odium.
And then trying to break the rules, a la Kaladin and the final boon at the end. Not knowing the rules and the precepts of the challenge is going to get some people burned. Probably Hoid. He's always a rule breaker, he'll try to find some loophole in the agreement between Honor and Odium and exploit it for his own ends.
Final outcome - I think whoever is Honor's champion is(Probably Dalinar) will realize that you can't actually kill Odium, he's too invested. Or that splintering him will cause more problems than not. Instead, they will use Honors boon to become the vessel for Odium, basically killing Rayse in order to contain Odium.