I picked up a lamp from a pile of free stuff by the side of the road. It needed a shade; while I thought about it, I put on a spare lampshade of entirely the wrong shape, and put it in its new home in our family room.
What I finally ended up doing for a shade was to follow the Japanese principle of building anything that is at high risk of being demolished via disaster from the simplest and most easily replaceable materials. I took a paper grocery bag, cut off the bottom and remove the handles, put it around a five-gallon bucket (the bag fit loosely around it), and then gave it two drippy coats of some spray-on fabric stiffener that someone had handed down to me.
With the fabric stiffener, the bag could hold its shape, and didn't need much of a lampshade frame to support it.
To hang the shade on the lamp, I did the simplest thing I could think of: take a dowel, cut it slightly longer than the diameter of the shade, then punch a hole on each side of the shade to run it through and across. (For this, I used a very beaten-up old dowel that I was about to throw out.) Then I lashed the center of the dowel to the top of the lamp's harp with wire, and slipped the shade over and onto the dowel.
The final result looks interesting with the lamp on, like very, very old parchment. The shade is a little short in proportion to the lamp, so I am thinking of adding something to the bottom edge.
UPDATE: I unravelled one side of a strip of burlap to make a fringe, which I stapled to the inside lower edge of the shade.
LONGEVITY UPDATE (3/2018): This lampshade frequently falls off its dowel, but it is easy to put back, and the fact that it falls down before it is torn down has helped this flimsy paper lampshade survive at kid level for ten whole months in my house. I have several times had to restaple some of the fringe; also a quick job.