This was one of those things that bugged me daily but I never did anything about it. I would kick around the seat, move it to the side, on top of the wastebasket, behind the toilet, etc. It was always in the way and there was no where it seemed to fit.
I took a shot that the handles on the side would fit with a Command Hook and bought a pack. Now my daughter hangs it up about 60% of the time after she is done and it is easily put away in a second by anyone who uses the bathroom next.