How do I place a light-weight point and shoot camera on a balloon filled with helium gas?
I hope it is obvious that the heavier side of the balloon will swivel to the bottom. Even if your camera is amazingly light, you will need a counterweight to balance that — something slightly heavier than the camera is the easiest. (You could also do several weights distributed in a balanced pattern. But either way: more weight!)
Once you have the counterweight, and the camera, the rest is reasonably obvious. This is going to be a fairly large balloon — not something you pick up at the florist. A standard balloon will only lift a few grams. Assuming you have something like a GoPro at around 200 grams, with the counterweight (for a total of 400g) you're going to need something like three-foot diameter balloon. If you have a heavier point and shoot camera, it will need to be even larger.
The ham radio club at the University of Hawaii has a nice page on balloon lift, which covers not just the amount of helium needed but different balloon types as well. (It doesn't cover mylar balloons, though, and I don't think you have many commercial options there, because even the so-called "jumbo" balloons are less than 30" in diameter inflated, and not actually spherical, so much, much too small.)
Once you've got that, though... use tape or whatever else to attach your camera and the counterweight, and there you go.