[GrussGott]

I'm no expert but I'd say you have to consider a few things: auto-focus, night shots and sound.

A decent video camera will have auto focus and that will make it way easy to "run and gun". Canon's Vixia are pretty good for cheap and panasonic has some great handy-cams too. The sony's have built-in projectors which is pretty cool but I hate their whole "buy these apps" after-the-fact thing. Anyway all these of those brands will make decent video cams that Auto-focus since your DSLR likely doesn't (or does it?)

Night shots will probably be better with a DSLR due to lenses, sensor size, etc you'll get better color saturation too ... so you could consider using your DSLR for that.

Sound is tough if you care - great sound makes great video but you may not need that if it's for kids and general use. If you do you can buy all kinds of shotgun or lavalier mics. If you're ok mixing in sound post-production the Zoom H1 makes for a great all-arounder that you can use for car video, etc

If it's purely for family use I'd be tempted to buy a sony handycam with built in projector.