You'll be way before the big crowds. There may be some snow but it should be ok. Yellowstone's website is very helpful
http://www.nps.gov/yell/index.htm . BRING WARM COATS. It's really cold at night.
Watch the 8 year old. Sadly, our national parks have been severely underfunded for the past many years, and are getting pretty rundown. We were last at Yellowstone in October and found a lot of the walkways in poor condition, which can be dangerous in thermal areas.
Do you know where you are going to stay? The village of West Yellowstone is nice but fills up. There are cheap motels and cabins and yuppie hotels. I'd recommend the Hibernation Station if it has space (
http://www.hibernationstation.com/) There's also an IMAX theatre in case the day is just too rainy.
I'd also suggest a cooler of food. Concessions may be open, but it's very pricey and frankly not very good. Have a picnic and watch out for Yogi :-)