A .243 would be OK, as long as you are up for another rifle when/if the child grows and wants to hunt Elk or Moose.
A 7x57, 7mm08, 6.5 CM or 6.5x55, even a .308 could be a lifetime rifle - with suitable cartridges loaded down for starters. There are some reduced recoil cartridges available commercially, if you are not a handloader.
Cartridge choice among that suite of chamberings is not that important, except for availability in the store. They will all kill game efficiently.
A youth stock, or altering the stock to fit is a requirement for comfortable shooting. Comfort and good experiences with a rifle is key to a youth shooting well.
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