I want to capture my young kids in good quality photos.

4/3 lenses are more common. Most optically perfect m39 screw mount lenses can be used perfectly for very low amounts (like $10 for a Russian-made Jupiter 8 50 mm f/2 lens) and four thirds sensors are bigger than nikon cx. this means better iso and depth of field, better pics. I'd go for Olympus which is also cheaper.

The J1 has too small of a sensor for high quality photos in anything but bright daylight. Nikon really goofed by trying to make an interchangeable lens system with a point and shoot sized sensor. The micro 4/3 sensor size will bring you better dynamic range, lower noise and more detail. When in doubt, go for the larger sensor. 

I'd go with the Olympus. First, it's cheaper. Also, the bigger sensor means nice depth of field (—I love the way family portraits look with a nice blurred out background). You'll also have access to a better variety of lenses and support with the Olympus—there are lots of micro 4/3rds lenses out there, and there are more coming. 

Whatever you choose… have fun. (:

Olympus epm1 is a good choice for this. Easy to use and great price and build quality.

Which do you recommend and why?

I'm loving my new Lumix ZS-15 for taking pictures of my toddler. It can handle movement and dim lighting very well. Great indoors and out. Very crisp photos, Great zoom. Small and light. And a good price.

checkout the samsung nx100 / nx200 series with the 18-55 lens. It's small, compact, very convenient and allows you to blow up prints to A4-A3 size posters if you like! :)

You should buy Nikon 1. Nikon says that this camera will capture pictures, even before you fully press the shutter button. And ... Nikon will work well with Nikon. Your husband already has a D7000. :)

see this

http://www.ephotozine.com/article/nikon-1-j1-vs-olympus-pen-mini-e-pm1-review-17564

I like the Nikon, has great image quality, I'd buy a j1