The .308 is a excellent all-around choice.

My pick would be a Ruger M77 Compact with a laminated stock...(my 7 year old son found it comfortable to hold and shoulder)...has about a 16 inch barrel...and most reports have found it very accurate. Sportsman's wearhouse had them for about $550, and they come with Ruger's factory scope rings.

If recoil is a problem, check out Remington's Manage Recoil loads...decent performance with about half the recoil! As she becomes more familiar with the rifle she can work up to full power factory loads or even Hornady's Light Magnum loads for very hot performance.