It was time to correct this problem, and Lego is always a great solution.
This is a simple stand, made of 14 pieces rummaged from the Lego boxes we've accumulated. The best part of it (other than it's Lego!) is the sloped pieces on the inside that make the phone slide into a stable position.
Photographing black Lego is hard, so I also built the stand in Sketchup to better show the pieces.
A 6x12 plate serves as the floor, with three front blocks to keep the phone from sliding off, and a wall behind for the phone to lean on. The slopes on the back wall cause the bottom of the phone to slide forward against the front wall, adding stability that wasn't there without them.