Why are children called “kids”?
Why not lambs, puppies, tadpoles, kittens, seedlings…
Ever get a whiff of a baby goat?
Ever seen how baby goats behave? It’s uncanny.
My big-ass dictionary ™ is not with me at the moment, but I’d venture to guess that it’s from the German ‘Kinder’, ‘child’. Perhaps it would be a better question to ask why baby goats are called ‘kids’. 
kid n. 1.[17C-late 18C] a child. 2. [early 19C] a child of either sex, esp. a juvenile thief, known as ‘the kid…’ (followed by their surname). 3. [19C+] a friend or fellow, often used in direct address. 4. [late 19C-1970’s] (US prison) a catamite. 5. [20C] (US) a young woman, used affectionally, esp. in direct address. 6. [1920’s+] (orig. US) as one’s kid, one’s younger sibling. [SE kid, a young goat]
Source: Cassell’s Dictionary of Slang.
Damn.
Having said all that, the OED gives a date of 1690 and an origin of low slang.
1690???
Wow. I thought it was coined in the fifties. I can’t imagine George Washington calling Jacky, Patsy,
Nelly and George, “the kids”.
1690???
Wow! I thought the term was coined in the fifties. I can’t imagine George Washington calling Jacky, Patsy,
Nelly and George, “the kids”.
Why are children called “kids” ?
Because it’s shorter than “Useless Eaters”. 
Now now, we’ll see who’s useless when you start needing someone to buy your “Oops I Crapped My Pants” for you.
I like this theory, especially as the German singular, das Kind is not so far off and the pronunciation is quite similar.
[sub]Sorry, but I have had this stuck in my head since I saw the question earlier in the day:[/sub]
[sup]…Kid my only kid
you look so small you’ve gone so quiet
I know you know what I’m about
I won’t deny it…[/sup]
Or we have an even earlier date from m-w.com:
I think that references the introduction of the word “kid” but not necessarily the usage questioned in the OP.
My old pastor used to say that it wasn’t on to call children kids, because we’re supposed to be sheep, not goats… and goats are associated with the devil… and all that jazz. But while it sounds like the kind of thing narrow-minded Christians might get themselves terribly hot and bothered about, I heard him use the phrase himself a few times, too. I mean, common sense will find a way. As Jeff Goldblum should have said in Jurassic Park, just before they nuked the island *a la **Aliens.