Free: Contests & Raffles.
I think Karl meant Ruger American... All his other points are spot on.. I would lean towards 7-08 or even the 6.5 Creedmor. The 6.5 is a fun gun to shoot.
Savage youth compact 243, only weighs 6 1/4 pounds,$230 after rebate at Cabelas right now:http://www.cabelas.com/product/shooting/firearms/centerfire-rifles/bolt-action-centerfire-rifles/pc/104792580/c/553829580/sc/105522480/i/105523380/savage-axis-compact-bolt-action-centerfire-rifles/1960559.uts?slotId=13
Lots of variables here like size of child and sensitivity to recoil. My daughter shot her first deer this year. She is small for her age (7) but she showed the drive, desire, and mental maturity to handle firearms safely and absorb the information to pass hunters ed.If size and recoil sensitivity are an issue .243 is the natural choice. Otherwise 7mm-08 with 120gr pills would be my first choice. River American compact is a very inexpensive and accurate gun for the money. I set one up for whitpirates son and was holding 3/4 MOA at 500 yards.With my daughter we shot heart sized balloons at 50 yards with her .22 using field positions. At the end of each session we would shoot her .243 2 or 3 times the same way. For practice we used a 58gr vmax over 22gr imr3031. VERY light recoil and was plenty accurate. Basically equivalent to a .223We also did target acquisition and dry fire drills at home in the living room. I have taken more than a few youth Hunters out for turkey and deer over the years and one thing I see the most with new Shooters is there an ability to find game in their scope. The drills really helped my daughter in that department.