I've got two kids and what I would add to the observations listed in this article are as follows (these apply to any app that is intended for kids, but also apps where a significant section of the userbase might be kids - for example, my four-year-old is quite adept at Plants vs. Zombies, and given the depth of strategy utilized in that game I don't feel bad when he plays it):
1. No part of the application should require that you can read in order to use it or navigate it.
2. Don't pop up dialogs that a child cannot understand. For example, I watched as my child, while using an app intended for children, pressed "OK" on a dialog that asked if he wanted to turn on push notifications. That's just ridiculous! He'll press "OK" on any dialog because he just wants it to go away.
3. This really applies to all applications, but it becomes very apparent when watching a child use an app: the most desirable menu items should be larger and/or differently coloured than the least desirable ones. I.e. "Play" should be a big, brightly coloured button, while "Settings" should be small and tucked away.
4. Don't make it easy to do destructive things like delete accounts. This might seem obvious, but I have more than once opened an app only to find that my progress in it had vanished because my child had deleted my account (Plants vs. Zombies makes this too easy, for example).
5. On any app that has the potential to be enjoyed by both kids and adults, consider providing a kid-friendly mode that makes the game easier.
6. I don't know how difficult this is from a development perspective, but if possible, make the app resistant to having non-active fingers touching the screen while active fingers are attempting to use it. Kids will often grasp devices, especially phones, in such a way that fingers from (say) their left hand are touching the screen, and on some apps this causes them to cease responding to touch events from their other (right) hand.
7. If your app produces revenue through advertising, it probably shouldn't be marketed at young kids. Kids will press on the advertising and will just get frustrated when they arrive on a webpage somewhere, and adults will eventually delete the app because we don't want our kids feeling frustrated.