I can give information regarding the Magic Kingdom, only.
You have to show up before opening. In MK, they seem to pick the Family of the Day from the first few families hanging around the turnstiles on the left side of the opening gates. Of course, you must be cheerful and in a good mood. Talk with the CMs around the turnstiles, etc.
If you are picked, you get free autograph books and pens for the kids in the family. You then take the old-style fire engine from Main Street to the ToonTown station (before opening). You meet all of the characters that will be getting on the train with you, and get to have a picture taken with all of the characters.
This was us in 2006. It was our last day of vacation. Note that the characters will be different every day, but usually includes Mickey, Minnie, Donald, and Goofy.
Then you ride in on the train behind Mickey and get to throw confetti off the balcony as the CMs declare the park open.
Finally you get some more pictures taken with the Mayor of Main Street, USA and with Mickey, Minnie, and Donald. You get a certificate for a free 5x7 which you can use on one of the photos that was taken. And you get a certificate saying you were Family of the Day.