How much does participation in online predictive challenges (DREAM, Kaggle, Innocentive) improve your resume?

At the very least, it shows you are actively interested in the subject. Participation can be enough to show you are more enthusiastic about the subject than other applicants. It is also helpful for entry level data scientist positions where experience isn't necessarily expected, but appreciated. If you have won any competitions, or if you are at the 'Master' Kaggle level, definitely include it.