This post is based on submission by Jennifer Rohr.
People who are bored often seek distractions to keep them occupied, which may lead to fewer opportunities to integrate your thoughts.
Scott Adams (of Dilbert fame) wrote an article in the Wall Street Journal suggesting that the current trend towards distraction is leading to a lack of boredom and, consequently, to a lack of creativity.
However, proneness to boredom has been linked to depression and other psychological issues, so it is probably not the fount of creativity. However, opportunities for introspection and reflection—like those found during meditation or daydreaming might be better explanations for creativity in low-stimulation environments.
The Heady Thrill of Having Nothing to Do at The Wall Street Journal for the full article.
Everyday attention lapses and memory failures at the University of Waterloo for research into the link between proneness to boredom and depression.
The Benefits of Boredom at SparkAction for a discussion about how structured activities for children can supplant their annual summer boredom.
Boredom - A lively history by Peter Toohey
Daydream at Wikipedia