Assuming the hunter will practice and become a proficient shot with the rifle at 100+ yards (my own measure is 4 shots consistently within a 6" diameter circle at the range desired), then a .222 or .223 will do the job WITH PREMIUM BULLETS. They will be fun to shoot, probably the safest in terms of avoiding bruises, and will take the game if the shooter does his/her job. If you feel a step up is in order to increase killing power, the .243, .250 Savage, .257 Roberts are all considerations. Keep the bullet weight as low as you can in these cartridges if you want less recoil.
.357 magnum is a possibility, but the range will be limited to about right at 100 yards; however, a huge advantage here is that they can practice with .38 special ammo and hunt with the real .357 as GB described above. The others mentioned above (6.5x55, .260, 7mm-08) are also nice beginner guns, but push heavier bullets unless you reload, and I would try to keep bullet weight around 100 grains max in the high velocity cartridges to reduce recoil. With the .357 mag at lower velocity you can use up to 158 gr without a problem.
A high-quality recoil pad like Pachmyr will also help a great deal to avoid bruises. Choose a gun with a good pad or upgrade if not, it is worth it in your case.
Good luck with your daughter's first hunting season, it is a joy you will remember fondly. I shared hunting season for the first time with my oldest daughter last year.
