There is a lot of food for thought in this question. I would like to differentiate a bit with respect to the character of the contests.
The subject of contests I know is fairly inconsequential. There are spelling bees, when it comes to mathematics, there are contests in symbolic integration, marshmallow eating contests and beauty pageants. You rarely hear about vaccination contests for nurses, sentencing contests for judges, or fire fighter competitions going beyond carrying buckets. As Victor has pointed out, computational science is a serious task, which requires skill, understanding, thought, and perseverance. All these cannot take effect in a short time contest.
What you find more commonly are coding challenges like the HPC Challenge mentioned in Pedro's comment. These take place over extended periods of time, and allow the contestants to use all the tools and information they need to produce a high quality result. They are usually sponsored by corporations (for instance Google) or other big entities because they need the result or use them as a recruiting device. But they have a lot of potential to raise interest in scientific computing in young people and to give them exposure when they succeed.
A third option are prizes for already existing accomplishments. Scientific awards have seen a considerable inflation during the last years, but by many are still seen as an important indicator for scientific achievement. Compared to all the medals that are awarded for instance in pure mathematics, computational science could definitely do with some more.