My kids love the Syma S107 toy helicopters, which cost just $20 and are amazingly stable. The big breakthrough for this generation of toy helicopters was the introduction of gyros, but I just took apart a broken one and was impressed to see that it has just ONE gyro (circled in black above), mounted vertically.
I know that the coaxial props create a natural mechanical stability in the XY direction, and I assume that gyro is stabilizing in the Z (yaw) direction. But is it really as simple as that? The stability is astounding. They're just motionless in the air until you budge the sticks.
BTW, there are no magic sensors on the bottom of the board, either: