Sorry, I'm not sure what you mean.
I'm thinking it would go something like this: users would be encouraged to track examples of contrarian contributions. At the end of the month, there would be a nomination process (with pointers to examples of contrary statements) and then voting on who was the best contrarian. (Whoever maximizes quality of statements degree of disagreement with other users at the time they wrote the statements. Number of contrary statements made could also be a multiplier, although that might be a bad idea if we want to avoid flooding LW with disagreeable contributions. Come to think of it, "contrarian *contribution of the month might be a better award".)
Allowing users to nominate themselves seems like a generally good idea, in case we are subconsciously avoiding our beliefs' real weak points, and to fight availability bias (individual users are more likely to remember good contrary comments they made early on in the month). There's probably no reason not to keep the registry open for nominations all month long.
If you're asking if there will be an award, maybe we could give them karma somehow? Personally, I suspect just winning the title will be a significant motivator.
An interesting variation would be to encourage established users to create alternate accounts to be contrary with, and only step out from behind the alternate account if they won the award.
One problem is quantifying the degree of disagreement. For instance, in one sense this recent discussion post of mine is very much in line with stereotyped opinions of what Less Wrong thinks, but in another sense, it got a substantial number of votes down (was negative for a good while after I created it) and the top-rated comment on it, voted much higher than the post itself, expresses disagreement. So was I being contrarian or not?
http://lesswrong.com/lw/bfy/you_only_live_once_a_reframing_of_working_towards/
Another idea is for contrary posts to specifically state that they are nominating themselves for the award within the body of the post. This could create a different dynamic when responding to the post, if it was explicitly pointed out that the post was something you might disagree with but might be correct anyway. (Probably not that good of an idea.)
I'm worried that LW doesn't have enough good contrarians and skeptics, people who disagree with us or like to find fault in every idea they see, but do so in a way that is often right and can change our minds when they are. I fear that when contrarians/skeptics join us but aren't "good enough", we tend to drive them away instead of improving them.
For example, I know a couple of people who occasionally had interesting ideas that were contrary to the local LW consensus, but were (or appeared to be) too confident in their ideas, both good and bad. Both people ended up being repeatedly downvoted and left our community a few months after they arrived. This must have happened more often than I have noticed (partly evidenced by the large number of comments/posts now marked as written by [deleted], sometimes with whole threads written entirely by deleted accounts). I feel that this is a waste that we should try to prevent (or at least think about how we might). So here are some ideas: