So, I don't really have any proof for this, but I was talking with a few other Sharders during school about how to shatter Shards and we came to this conclusion.
Shards must be shattered by forcing them to act against their primary intent. It clearly doesn't work to just kill the vessel (examples of this is Mistborn and RoW), so there has to be another way to shatter the shard. Basically, the only thing that I think could truly break a power is forcing that power to do something it cannot. In this case, that is act against its primary intent.
In the case of Honor, Odium probably managed to get Honor to break an oath. The Stormfather said that Honor got to the point at the end where he didn't care what the oath was about, as long as it was kept. Odium could have used this to force Honor into a Kaladin-in-Words-of-Radiance situation that forced Honor to break an oath. This would go against everything the power stood for, effectively destroying it.
Like I said, I don't really have any proof, and I don't think that I did a great job explaining it, but hopefully it makes sense!