That is a great question, Jon. I mean, a helmet would seem to be an essential safety tool, no? They sure have saved MY tender brain-case on more than one occasion. So what is the deal with huge price discrepancies? If you pay less, are you getting less safety?
The answer is: No. At least, not if you do just a tiny bit of homework. You need to make sure the helmet has CPSC (Consumer Product Safety Commission) or ASTM certification. That's it! With those certifications, the helmet has been shown to meet essential safety requirements. You don't spend twice as much and get twice the protection. What you typically buy as the price goes up is less weight, maybe more style, and better ventilation made possible by more integrate engineering (hence more $).
That's said, I'm not sure how many $10 helmets there are out there, or whether a helmet at that price would meet CPSC standards. More like $20 and up, for something like a Bell Rex ($22), which has kid-friendly graphics and a really good strap system. Giro's Rascal has cool-looking flame graphics, as well as integrated LED lights for visibility. It's $40.
Helmets typically allow for adjustment of the sizing, so one should last at least a year. Also, look for sales at bike shops—you often can get last year's model at big discounts. I'm not aware of any upgrade offers as fit changes. If blue jeans makers don't offer that, then helmet makers won't either. Kids grow.
The most important thing in any event, is PROPER USE. I rarely see little kids wearing their helmets properly. First, straps should be snug. Not tight, but snug. They should form a Y beneath each ear, and the buckle should centered under the jaw. You should not be able to easily pull the helmet off their head. Secondly, the helmet MUST BE LEVEL. Not cocked back on the head. If the forehead is visible, then it's not on right.
So there you go.