I bought my son a Rem 700 7mm-08 when he was 6yrs old. He is now 10 and has 8 Texas whitetails and several hogs to his name.
I have been loading it with reduced 120 grain ballistic tip loads and will slowly work him up to full strength loads as he grows.
I had the original tupperware SPS stock cut down to 12.5" when he first started shooting but have since dropped it into a B&C stock that is 13" LOP and he shoots it just fine.
If your not a handloader Remington makes a reduced recoil load that is really easy to shoot.
Good Luck!
That's what I did too. A 700 ss 7mm-08 with a few inches cut off the tube and the stock swapped for a youth model with Supercell pad. My girl killed her first deer at 7 with it. Reduced loads kill just fine inside 150yds.
Bob's mention of the 6.5 CM is a good one too. A 260 would also serve.