I'd search locally at the race track. Unlike street/dirt/etc bikes, race bmx bikes are generally tailored specifically for kid's size. IE the JR's really are JR's etc.
The bikes are ultra light and pretty durable, but of course not near what a heavier bike would be.
You could most likely pick up a used bike for dirt cheap as well, kids grow... fast, and when bmx racing, parents tend to constantly up bike size until the kids size slows down a bit.
If you want new, there are plenty of entry level priced bikes. In the end, I would definitely stay away from heavy bikes unless she's interested in dirt jumping (which you can still do on a race bike).
Just my thoughts on it
A bmx track is also a great place to learn all sorts of skills on a bike as well, ntm etching safety into their head at the same time.