Shoot the Margins
Shooting on the margins often leads to the most fun, funny, and memorable moments of family video. By "shoot the margins," I mean those moments before and after the "meaty" part of the video. Another way of saying it would be, simply, "Don't turn the camera off yet!"
For example, you might capture all of your child's excellent piano recital on video, but the best moment of all is his expression of relief when it's over. Or when your daughter successfully blows out her birthday candles, you might be tempted to put the camera down, but if you just leave it running a few more minutes, you'll catch her shoving her whole arm in the cake. It's no surprise that those fun moments happen just after the "real" action, because all subjects tend to let down their guard and allow their natural, non-photographic emotions to take over. Those are the moments you don't want to miss on video.
The iPhone is great for shooting the margins because it's light enough to hold for awhile—and footage can easily be tossed if that minimal extra effort didn't pan the proverbial video "gold." The only trick is to remember that if you're holding the camera for some time, you need to remain steady and continue to monitor your framing.