Pop songwriters talk about finding the hook in a song, the hook that gets people wanting to hear the song over and over again.
It's there in good kid's music as well, obviously, just adjusted for audience. And while a good song is always important, the hook in kids music isn't always a melody or chorus.
One thing I've noticed in seeing how kids react to a wide variety of music, and in making music for young kids, is that great songs often have the part where. Kids will say, I like the part where we make the elephant noise, or I like the part where she sings up, up, up, DOWN and that funny whistle comes on, or I like the part where we have to find the different colors. Even very young kids who aren't really talkers yet will clearly show you the parts they like in their favorite songs by how they bounce, smile, clap, etc.
So when you are evaluating music for your young kids, or when you are making music for young children, ask yourself, "What's the catch...where's the hook...what is the part where...". It doesn't have to be just one, because, of course, different kids will latch on to different parts...but you should be able to spot potential parts that kids will love and want to hear again and again. It's not always possible to guess which parts kids will love (kind of like you never know if a kid is going to like the box their birthday present comes in more than the present), but you should be able to identify if the artist is thinking in those terms...is thinking of what will be fun for a kid to do, sing, hum, etc.
I think this is one thing Laurie Berkner does that has helped make her so popular. Of course she has a fun voice and is great musician who writes great melodies, but her songs almost always have parts, little actions or sounds or verbal combinations that a kid, who is already enjoying the music, will hear and say, "I like the part where..." And those parts make her really nice songs more than something kids like, but something they want to hear and experience over and over and over.
Look for those parts and notice which parts your kids react to...it will give you a great insight on planning some fun acitivities they'll enjoy.