I like Denali's ideas best, cuz I'm more of the engineering type. By the way, that would be a good question for Shapeways' new Ask An Engineer video blog. Can a living hinge be made? What material comes closest to the properties of polypropylene and other similar plastics?
Though a super simple contest like you thought of Bart would be really fun, I cannot readily think of any simplified ideas at this time. Simple and elegant, perfect for a weekend of fun. Makes me think of the Eight queens puzzle, simple and fun. All I can think of along this line would be something like designing a device that uses a predefined magnet to do a predefined task.
The ideas I have are way too complex for a weekend I think, but still, these are a sampling of the kinds of things I think about every day that I think would be fun as contests.
Design the ultimate printability demonstration model. One of the biggest problems with 3D printing is getting to know and conforming to the design rules for each materiel. I'm thinking of a demonstration model that could be devised and printed that shows what can and cannot be readily done and then make it downloadable so people can use it for reference.
Using free software, find the best way to make a hollow shell of a predefined design.
Design a rocket that uses a predefined rocket motor that is completely 3D printed, including the recovery mechanisms.
A zarf for the modern age. Design a zarf that uses a predefined ceramic model as the cup.
Design a 3D printed flying machine using a predefined rubber-band.
Design a device that keeps time.
Design a device that uses hydraulics to preform a particular task.
Design a device that measures weight.
Design a mechanical computer.
Design a device that crosses a gap by its own self contained means.
As for Glenn's multiple materials idea, design solar powered ANYTHING that uses multiple materials.
James