manicmike

I think Sewnoma's idea is a better one. Introducing a little girl to sewing with plastic rubbish that won't make a decent stitch could easily put her off sewing permanently. My sister was one of those (although she was an adult at the time): Bought a Brother, struggled for years to make it work, left it in the cupboard and never sewed again. I've met quite a few people who shared this story. Older machines (I'm thinking pre-1965) have a quality that makes them pleasant to use. They can be understood, fully controlled, self-maintained, they won't break easily and with maintenance will last for hundreds of years.
I'd recommend a 401 or 411 for a beginner. Here they can be found reasonably cheaply (last one I paid $25 for - after a service it looks and goes like new), and can do just about everything. Only drawback, if you could even say that, is that the slant attachments are slightly harder to find. The machines have no electronics at all, are not that heavy (although they are reasonably heavy) and very solidly built.
Electrical wiring is a possible issue with older machines, but if you get it checked by an OSMG (old sewing machine guy) or even buy it from one you should be OK. I haven't seen a 400 series machine yet that had an electrical problem but it's not worth the risk.
With a bigger budget (or some luck) a featherweight would be fantastic for a child. Very light, very cute (they look like a child's machine) and produce an excellent stitch. I considered getting one for my girl (who's 12 now) but instead got her a 1970 Singer 498, which she's still scared of (she's hand sewing nearly everything until now, but uses the 498 when she has a big sewing job). The 498 cost $7.50 plus $32.50 for the replacement nylon timing gears. Those gears are only in post 1964 Singers, and not in all of them.
You have plenty of options but I guess it comes back to there not being any old machines in your area. Maybe a day trip out of your area with the little girl is in order.