This was originally intended for the toss-it contest, but it didn't quite fit, I got hung up on the video, and they the local virus swept through the family and I never got around to finishing it. Now that the toss-it contest is over, I can just publish without the video, and maybe it will still be interesting...
This isn't so much plans for a particular paper airplane, as a concept and technique that could be used for a whole new style of airplane, not TOO much more complex or expensive than the typical all-paper airplane.
Step 1: Weighing Paper
Here, we have a regular 8.5x11 inch piece of paper. It weighs about 4.6 grams (Isn't it fun to mix those metric and american units?) If we fold this into a paper airplane, it will still weigh the same (of course), but only PART of the weight will be used in "functional" parts of the plane like the wing surfaces. In fact, in a model like the classic Dart, only about half of the paper will end up in the wings (and a good part of that is folded in multiple layers, so it's not particularly efficiently applied to the wings.
Step 2: Weighing Other Things.
So what else is shaped like the fuselage (body) of an airplane? How about one of those long, skinny balloons that are typically twisted into animal shapes?
One of these surprisingly rigid when inflated, at least compared to a piece of paper. You can attach wings and a tail with tape, weight the nose, and create a number of interesting plane designs. And it's still cheap and still easy to experiment with!
Here we have some balloons, weight in at 1.6 to 2.1 grams. Slightly more when inflated, of course (thereby demonstrating that air has mass!) That may not seem like a lot less than the paper when you're looking at the un-inflated balloon, but that 2.1g balloon is about five feet long when inflated. That's either a very large plane, or ... we can use PART of the balloon!