Of the 3 you listed, .357 would be the most preferable - if I was buying a new barrel for my 7YO, that's what I would choose. It has good performance, has low recoil and is relatively inexpensive for shooting practice.
I don't have any experience with the Hornet but from what I have seen, leave it to varmint hunting. The .223 is reported to be a decent deer round when loaded up with such bullets as the 60gr Partitions and the 64gr Win SP, both of which I am getting ready to work up for my T/C Super 16 Carbine (I
MAY let my 7YO use that next season, as long as we get some good range time; where I hunt, he will be limited to 50 yard or closer shots with whatever he shoots, so I am not worried about long range performance). Of course,
even better, I just picked up a used Handi .30/30 barrel and am looking at loading that up with some 125gr BTs or Rem PSPs for him to use instead (I'll have to see how it recoils first). Good thing I have a few more months to figure that all out.