Family History isn't ALL about remembering history, its about creating it as well.
There are a lot of great ways to document your history, such as scrapbooking and journaling, or blogging.
There was a great article in The Friend last month about journals. "Journals from the Past" which highlighted a couple of pioneer journals and what we can learn about journaling from those pioneers.
You don't have to be able to write to keep a journal. Younger children can either dictate to you (or an older child) what to write, or simply draw a picture about something that happened to them. Journaling is a great down/quiet-time activity, is appropriate for the Sabbath, and is easily done on the car during long road trips.
You may also want to keep a family journal and let all members contribute, either by writing entries themselves, or by sharing in a family-council type forum what they feel was important enough to document.
But remember that nothing is too trivial to track. The things you think are mundane, the weather, your daily routine, etc. will be fascinating to someone 100 years from now, or even 20 years from now.
If you haven't seen the "Kid History" videos on youtube, I recommend checking them out.
Embarrassingly enough, I had enjoyed several episodes, but had not thought about the family history value in them until I caught an on-line article about them. You wouldn't have to go to all the trouble they do, but videos are always a fun way to document history, and nothing says historic reenactments have to be reserved only for bringing back past generations.