There is an entire subgenre of children's music meant to appeal first and foremost to the parents. Particularly when it comes to baby and early toddler music. Kids that age don't really have peer groups they're trying to fit into, and very little in the way of preferences. So, if mom and dad are going to listen kids' songs, why not at least make it enjoyable?
Children's music is also a great place to reconnect with mediocre pop and soft rock artists from fifteen years ago. The gold standard here is probably Caspar Babypants, who makes grownup music as the lead singer for Presidents of the United States of America (don't lie, you like "Lump" and "Peaches" when they came out), but in his spare time has some hilarious kids' albums. Mr. Babypants is a local celebrity in Seattle, making regular appearances at public libraries and Top Pot Doughnuts.
Other successful nineties acts who've expanded into children's music include Lisa Loeb and They Might Be Giants.
Frances England, who I had never heard of pre-baby, has lots of fun original tunes plus a few classic tracks. She's fast becoming a favorite in our house, with tunes like "Bicycle, Bicycle", "Cookies & Milk", and "Jacques Cousteau".
Adult artists performing genuine kid songs (think "Wheels on the bus") are harder to find. The Muppets Green Album is pretty good. The For the Kids series has a few decent tracks.
Last but not least, reconnecting with the children's music of your childhood era can be fun. As an added bonus, you'll already know the words to many songs.
The rise of free and cheap online music makes it super easy to sample this stuff. Give it a try!