One idea I only had a line or two to suggest, was to have your children make a movie. I've never actually done this with children, but for dramatic kids, or techy ones or those who like telling stories, this may be a fun activity.
If your family already has a video camera, and a PC or Mac, they have what they need to make a film. Just add some imagination and some effort that hopefully they will regard as play, and you may set the next budding Spielberg on his or her path to greatness.
Films are showing up at Sundance made from the same ingredients, so having kids make a movie at least the family will want to see is quite doable.
What you really want to do, is try and get children to think about a story first and then figure out how to put it into a video. You'll need slightly older kids to pull this off the way I'm going to suggest.
The goal is to see if you can spark an interest in your kids. If you do, they can themselves do longer, more involved productions after.
For the first film, the parents may need to act as the Producer and Director. Hopefully if your kid enjoys the process, they will step up and take the reins for the next production.
First thing to do is to come up with a story. Maybe they will want to write their own, or create a fan fiction around characters they know from stories or films. As the producer/parent you want to keep the story small and reasonable. You want to keep videos to less than ten minutes so you can post them to YouTube. Besides, it can take hours of work to produce a minute of film, if you do it properly. So keep that first film short.
In fact, for the first film, just to show them the techniques, I as the parent would record say two minutes of some program my kids like to watch, and suggest they recreate the scene. When we learn handwriting, we don't sit down and try and write our own story, we copy printing we see before us. To learn the handwriting of film, let's copy a brief scene.
For sake of argument, lets say it is a brief scene of Hannah Montana and her friend Lily (or Oliver depending on your actors) just talking.
Write down the dialog and actions from the scene. Don't worry now about writing it in a formal script format.
They will expect that now they will leap to their feet while you point the camera at them and they run through the dialog. Go ahead and do that. When it's done, explain the video isn't done.
Go back to the video you recorded from the TV program and show them how there are a lot more shots in it then just the shot of the two of them. Each actor had closeups. And there may be over the shoulder shots and more.
List every shot that is used. Then use index cards to draw out the shots. For simplicity sake, let's say the scene just had a two shot, and a closeup of each actor both as they said their line, and reacted to what the other was saying.
Setup a close up on one character, and have the two actors run their scene again. Then setup on the other actor and do the scene for a third and final time.
The final process is to edit the scene. Microsoft's Movie Maker on Windows or iMovie on Mac are free or inexpensive video editing software. For this first movie, cut it together just like the sample scene as you teach your young filmmaker how to edit.
Make sure you put in credits with their name. Nothing will excite them like seeing the name on the screen.
Did they enjoy the process? If no, well, you tried to expose them to something new. May be you planted a seed that will erupt later. At least now when they watch a film, they will understand some of what went into it.
If they did enjoy making their film, review with them your simple process.
Explain to them movies often begin with a treatment. A treatment is a brief, often one page, telling of the over all story.
From that little acorn, a script is written. There are scripts to real movies they may have seen available on the net if they want to see how one is written, and write their future scripts in the proper format.
Explain to them the designing of each shot and drawing them on cards is called Storyboarding.
If you truly light a fire under them, you may want to consider purchasing the book "The DV Rebel's Guide: An All-Digital Approach to Making Killer Action Movies on the Cheap" by Stu Maschwitz. I haven't read it yet; it is on my own Amazon wish list. It's everything the budding filmmaker needs to getting started.
Throw the film up on Youtube.com, and point friends and family to it. Then ask your budding Billy Wilder, "What's your next film going to be about?"