If you're looking for a microphone for a child, you'll probably want one that's robust, looks good and doesn't cost too much money. Bear in mind, however, that there's a difference between microphones designed for live and studio use.
Vocal microphone for Children
A microphone for live use is usually more robust and designed to be held in the hand, which makes these types of dynamic microphones a good choice for children. Investing in a microphone stand can be a good idea too. It's best to choose a mic with a jack or XLR connector so that you can easily plug it into a suitable active speaker or battery-powered speaker.
Microphones for recording
If your child wants to record their vocal performances, choosing a microphone with a USB connector can be a good idea as these can easily be plugged straight into a computer. A dynamic microphone can be used for recording too, but generally speaking, using a studio condenser microphone results in a warmer and more natural sound. Condenser microphones are more sensitive which means it's better not to hold them when recording.
