My first deer/hunting rifle was a Browning BAR in .270. Heavy, but a great gun. Shouldn't have sold it.
I would recommend a M70 Featherweight in a suitable caliber for your youngster. I think 13 is old enough to handle a .308, .270 or similar. A CZ 550 American in 6.5x55 SE would also be a good choice - if you can find one. IMO, I would want to get him a rifle he could hold onto for his lifetime and not one that he would feel like he needed to replace sometime down the line. Raise your budget a bit and have him contribute as well so that he has some 'skin in the game,' so to speak. Then he'll cherish that M70, the Rifleman's Rifle.
l told my pap and mam I was going to be a mountain man; acted like they was gut-shot. Make your life go here. Here's where the peoples is. Mother Gue, I says, the Rocky Mountains is the marrow of the world, and by God, I was right.
- Del Gue