For your inexpensive binoculars most will say Vortex, DiamondBacks are just above your $200 budget, but better binos than the Crossfires, for the few extra $$ I'd say for the kid it's worth the money.
As far as the numbers go for power and objective, I always go with 8x42, this allows you with 5.5mm of light (low light viewing) as far as 10X goes to get the same performance you would need a 52mm Objective and now you are talking a much larger binocular. 10X vs 8X... 10's will be harder to hold "still" and the slight increase in power in minimal in what you see. 8X at 80yards you view like 10yards away, 10X at 80yards view like 8yards... a 2yard difference at 80yards, that's 6 feet. 8's will always perform better in low light situations. Even at longer ranges the 8's shine, just add a zero to those numbers. 8X at 800yards you view like 100yards away, 10X at 800yards view like 80yards... a 20yard difference at 800yards, that's 60 feet.
Anything past 10X were talking spotting scopes.
Personally, I own Swarovski 8.5x42 EL's and 10x30 SLC (I found a great deal and couldn't pass them up) and my 8x42 Alaskan Guides, I have a few other cheapos but rarely if ever used... Nothing can compare to high-end precision optics, granted way beyond your price point but truly outstanding optics.
MK M GOBL