If the recipient will be backpacking 10+ miles over hill and dale, then weight is the most important consideration. Look at 8x25 compacts and compare weights, they vary considerably. Small sizes are easier for older children to handle. (There are plastic children's binoculars for the little ones.)
If you're backpacking 2 miles from Starved Rock parking lot to Lover's Leap, over plank trails with staircases on the steep hills, then you have more latitude or should I say longitude? 8x42 is good for general use, and 7x50 would be good for hunting as noted. 10x50 is too heavy for most of us to carry far from the car, and too wobbly without some kind of support.
Athlon, Bushnell, Carson, Celestron, Nikon, Pentax, Vortex Diamondback have ok reputations. I've been happy with Pentax. If you need to downsize to meet your budget, 8x32 is a good compromise.