this question comes up alot, not only on here but i hear it local too. i tell everyone the same thing, first ask yourself a few questions(BE COMPLETELY HONEST,THE CHILD WILL BE THE ONE WHO SUFFERS NOT YOU) sry about caps but some people need reminded.
How much has the child shot? if at all what? 22lr,410,20ga,etc
How much practice will they get before season?
Do you have anything that you can have them practice with before purchasing a special gun for them?
Do you handload?
What type of terrain do you plan on hunting with them? sitting,hiking,etc?
these are basic questions i use to make a decision there are more but you get the idea.
ok,lets check options(with my opinion to follow)
357mag- i feel this is a bit low on power for a kid,recoil is next to nothing
357max- i have no experience but i hear good things,remember you'll have to have a 357mag reamed in a handi
243win- you need to be a very good shot to use this correctly,recoil is a sharp kick,very loud with lots of muzzle flash
30-30-you can't argue its effectiveness,recoil is more of a push than a kick
308win- might be good if you need more range,recoil is going to be on the high end for a kid
44mag- good option if range will be 100yds or less,recoil is manageable
there are more options for the handloader,if you need more input let us know,i'll be glad to add to your confusion