I got my daughter a Crickett for her first .22 when she was five. Chipmunk rifles are very similar to the Crickett and another good option. I liked the Crickett because of its simple design, my daughter liked it because I let her pick out which one she wanted. With all the different colors of laminated stocks a kid can pick out their own special color and it makes the rifle more special to them. They sell as well adult sized stocks for when the kids need them for a reasonable price. I don't recommend getting the scope for it, I feel kids need to learn to shoot with sights first.
You don't need to buy a super expensive .22 lr for kids as they will outgrow it very quickly. My daughter is now ten and I needed to buy a bigger stock for it a couple of years ago for her. The problem was when it became too small for her, she didn't quite fit the adult stock. Now that she does I asked her if she wanted to buy a new stock for it and she said she wanted to keep it like it is because it is her first rifle and special to her. She is happy to shoot my .22's now and her favorite now is a .223 Stevens 200 I gave her.