If the hat is for an outdoor activity such as hiking or biking, a wide-brimmed hat will protect the girl's head from the sun and wind. A baseball cap or a bucket hat is a good option for this situation. For colder weather, a beanie or a winter hat will keep the girl's head warm.
If the hat is for indoor use, like at a ball game or a school play, a smaller, more fitted hat will be better. A baseball cap, a fedora, or a beanie will all work well for this situation.