Jazzoctopus, if you're reading this, comment, please, on the possible link between "kinder" in German and "kid" in English.

Hmm. . . I can't profess to be an expert at the relationships between German and English, but I can try to help.

In German, child is "das Kind", which, because of the language similarities, could feasibly be linked to both child and kid. Perhaps "kid" started being used to refer to children when the US began using Kindergartens.