Starting on synth/keyboard vs piano (for a kid)

[Bartimaeus]

I'd say wait a month or two, and if she's still into it then get a Yamaha P-series keyboard. Ideally you can find a used P-series with local pickup in your area, craiglist can be good for this. It's close enough in feel to a real piano that practicing on it will carry over if she wants to learn real piano, but it also has MIDI and can be resold without too much trouble. Plus it has a headphones option unlike real pianos (a lot of classical musicians use one to practice for that reason).

You could think about weighted keys as acoustic vs electric guitar (although I think it matters more for keyboard). If you start on an electric guitar, switching to acoustic is a challenge (though not a huge deal). But if you start on acoustic guitar, switching to electric just makes everything feel easier. It's better to start with weighted, even if you only end up using synths down the road.

You also shouldn't discount the effect that having something nice can have on inspiring someone to play. Some keyboards will sound bad no matter how good of a player you are, and they can end up discouraging you.