Everyone likes free stuff; it is no different for your customers. If you are having some trouble engaging customers, getting conversions, or increasing exposure, a giveaway/contest might provide some fireworks.
People will do some amazing things for just the chance at free goods or the chance to feel important. For example, if you are looking for a logo redesign you might tap the creativity of your website visitors. You will find a number of talented people striving to impress you and their friends with their submissions.
In addition, a giveaway or contest can inspire some great brand loyalty if you carefully manage your relationships with the contestants. In addition, allowing a public vote will encourage the contestants to bring in their social networks to support them. This brings in more traffic to your site and gets great exposure in a myriad of places.
Staying in touch with the people you inspired with your contest will leave you with loyal contacts that identify with your business and may again contribute in the future.