Me and my boyfriend were picked to be Family of the Day at DHS in September 2007 and at Epcot in May 2010. Both times we were the first people to be at the turnstiles, however that didn't matter as much at Epcot. Both times we were really excited and happy + we were wearing Disney shirts.
At DHS, we got picked at the turnstiles after talking to a jolly, older CM. This was during the Year of a Millions Dreams, and I don't think they pick people to do this every day anymore.

It hasn't happened each time we've been since then, anyways. The CM that picked us recorded everything on our camcorder while we were in the show! We got to ride in the pretty white car (which, I'm not sure if I really believe it or not, but we were told it was one of Walt's cars!) down Sunset Boulevard down to the Tower of Terror. We got to ride by ourselves on the first ride of the day, were given a free souvenir photo, and were given Family of the Day pins. The CM let us pick up our photo, as well as our Year of a Million Dreams certificate, at Guest Services later that day. He also included a very nice, hand-written note that wished us a happy vacation

This is us behind the rope at DHS:
At Epcot, we got to the rope in front of the fountain and were asked if we were excited about our day, etc. After a few more questions ("Is it just the two of you?" being asked multiple times, so I had a good idea about what was about to happen) the CM asked us if we'd like to help Mickey and his pals open the park! Duh. So she let us behind the rope and we went to stand by the fountain. We had people explaining what was going on and then Mickey and everyone came out and took a bunch of pictures with us! (Photopass as well as our own camera.) Once the little show was over, we got into the Test Track car and rode to Test Track. We got to ride twice by ourselves and were given a free photo at the end. We were then asked if we'd like to ride Mission: Space. Duh (again). So the CM that had been with us all along walked us to M:S and took us to the "control center" behind the glass of the queue. We got to push all the fake buttons we wanted and were told to sign the guest book that Family of the Day members sign! We were then taken to the front of the line. Once the ride was over, we were given two vouchers that were basically paper Fastpasses that were good for up to four people for any ride at any time that day. (I kept one as a souvenir

) We were also offered VIP seating for Illuminations that night but we had to be somewhere else

I hated not getting to do that, because Illuminations is my absolute favorite fireworks show. Here's some pictures from our morning!:
We also have both of these experiences recorded on our camcorder. I wish I could upload them, but I have an old camera and I don't know of any way to get them on my computer.
Getting to open a park is a truly magical experience! I feel so lucky to have been picked twice. Now we just have to conquer AK and then have some kids so we can open MK!