We were chosen to be "family of the day" on the Magic once. We had an early port arrival time. 10:30, if I recall correctly. The three of us all wore the same red
Disney Cruise Line t-shirt that you can order from the
Disney Store website. Our son happened to be turning 4 years old on embarkation day, but that was probably just a coincidence. After checking in at the desk, we lined up for a welcome aboard photo with Minnie and a cast member who had been watching us approached us after the picture and asked if we would like to be the "family of the day." We said sure and she asked that we stick around a general area so that she could easily find us when it was time to start the boarding process.
From what I could tell, it was just a random selection. But, as PrincessShmoo pointed out, it probably helped that we were wearing matching shirts. So did the fact that our child was happy and exuberant at the time. And, seeing that the cast member had other duties to attend to until embarkation started, I imagine our early port arrival time weighed in our favor on that particular day as well.