Besides Utah, another good bet is the higher elevation ski areas in CO. In recent years I've had some great April skiing at places like Loveland, Copper, Keystone, Arapaho Basin. For those places and the Utah resorts Bushwacker mentions the time frame you are talking about is great for kid-friendly weather, low crowds/prices, with usually near 100% terrain still open. You're lucky, for those type of western resorts it's a much better time for family skiing then Christmas or President's Week.
See here: http://www.dcski.com/articles/view_article.php?article_id=1212&mode=headlines
I've also had some good late Mar/early Apr skiing in northern New England and Quebec, but that is a little more hit or miss.
PS: that six year old sounds like a gamer