As mentioned before, paper is a excellent medium for engineering projects because
1. It's cheap and cheap is good for experimentation
2. Paper has a number of interesting and useful properties from a material science standpoint
3. From a pedagogical standpoint, these properties take on extra value because the material is so exceptionally ordinary. We want students to get in the habit of looking for the interesting in uninteresting places.
My recent fascination with the subject started with this idea for a class project/competition:
The task: build a platform at least 6" tall to support a 1' by 1' board (or piece of cardboard) and some weights that will be placed on it. (best of three tries).
Materials: 20 sheets of standard typing paper and 20 1" pieces of Scotch tape.
Objective: to support the most weight possible
In addition to the previously mentioned benefits of using paper for manipulatives, this project has a few other big selling points.
The students will probably be surprised at how strong an arrangement of paper tubes and cones can be and this surprise might help feed their curiosity.
The project is largely open-ended yet it has a well-defined metric of success. The better tower is the one that supports the most weight. There's little question about whether one design is better than another.
The project can employ some surprisingly sophisticated engineering.
The project can be used, with little or no modification, for students ranging from fourth graders to first year engineering students. There is enough potential for discovery that a ten year old and a twenty year could both learn something from giving it a try.
Two variations to this project immediately suggest themselves.
First is to use a fixed weight (let's say five pounds) and make the winner the highest structure that can support the weight.
Second is to use some function of weight and height. I'll let you think about that one on your own.