They Might Be Giants have been my favorite band for almost two decades, and they are, for the most part, extremely family friendly. They've come out with several "children's" albums, but I never saw the need...why a special album for children, when some of their songs are "Dr. Worm," "The Sun is a Mass of Incandescent Gas" and "Particle Man."
Honestly, I think everyone could love TMBG if they gave them a chance. They are most known in pop culture these days for doing the theme song to both "Malcolm in the Middle" and "The Oblongs," but I was listening to them long before that, and to me, those are their "sell out" songs.
Some of MY personal favorites of theirs are "Mr. Me," "Nightgown of the Sullen Moon," "They'll Need a Crane," and "Ana Ng." Check them out on youtube, see if you like them...I think that's a good mix of what they sound like. My children love their songs.
Other than that, Disney puts out a "Children's Favorite" series...I think there are 4 volumes...and Vol. 4 happens to be one of my favorite CDs. :) It's got songs like "Reuben and Rachel," "Jimmy Crack Corn," "Bicycle Built for Two," "Dan Tucker" and other folk songs.
My children also like Smoky Robinson, most oldies from the 50s, Arron Neville (sp?), Neil Diamond, Paul Simon, Marty Robbins (check out the song "Big Iron" while you're on youtube, we LOVE it!) and a lot of other old music.
***I had to edit, because someone above mentioned Me First and the Gimme Gimmes. They are an awesome ska band from my youth and they are excellent!!! If you like a punk feel and some brass, check them out. :)