The Mrs and I were having a discussion about teaching children to shoot and what kind of rifle is good to start them out on. Teaching a youngster the pistol is another subject for another post so I'm going to just stick to rifles for the nonce.
She likes the Cricket .22 rifle.
It's not a bad choice. It's perfectly sufficient to teach the fundamentals of shooting but I'm not convinced that a single-shot rifle is the best option. Loading single rounds can be tedious. Besides, you're there to learn to shoot, not learn to load.
I learned on a Remington Model 34.
It's a tube-fed, bolt-action .22LR rifle. While I understand the idea of not handing a semi-auto to a youth shooter, I think that the safety aspects of a single-shot rifle are over-rated. To reload a magazine-fed bolt action requires the same steps to reload as a single-shot but deletes the pain of having to fumble around with single rounds. Depending on the age of the new shooter a box magazine rifle might be a bit too difficult but reloading a tube action is much the same as loading a Daisy BB gun.
Granted, the Remmy M34 can be kind of hard to come by but Marlin does make the XT-22TR series and a few companies make a lever-action .22LR which have many of the same benefits.