Wait for a year on "The Moon is a Harsh Mistress". It is a wonderful, wonderful book, but at her age she might get distracted by the weird marriages (clan, chain, line, and other polygamous structures) and the light sex. Read it for yourself, if you haven't, and decide when she'll be ready.
I would urge caution on "Starship Troopers" but... that one's only going to instill a rabid love of those who put their bodies between us civvies and danger, so what's the harm? And "The Star Beast" is great. It's still a favorite of mine... has some of the best characters. Lummox, and John Thomas Stuart the whateverth, Mr. Kiku and Beulah Murgatroyd and Piggy-Wiggy... "Have SpaceSuit" is good, it might just inspire her to start learning on her own (it did me!)... oh, you have good taste.
Ender's Game is recommended because one of the messages - if you're ignored, what you do is make them unable to ignore you by being the best there is - is so true for a troubled kid. And she'll never, ever wonder if there's an "up" in space. "The enemy's gate is down".
I've read those Heinlein books. Citizen of the Galaxy was just always my favorite of them. Something about spy tales I always love. That book had me from the very first! That was why I chose that book to introduce the niece to Heinlein. Frankly, as you can probably guess, it just chaps my hide that they don't get Tolkien. That's okay, I think this niece will be wanting that "forbidden fruit" soon! :-)
I like the idea of giving her the Lovecraft book too. That way she'll get a mix of authors. This whole thing started because she'd visited me a few days 3 summers ago and discovered my delightfully varied library. At that time we read Island of the Blue Dolphins and she loved that as well. I am planning to keep introducing her to new authors and different genres and styles as she becomes older and if she stays interested. The only reason the Christmas gift turned into so many Heinleins was that I got a really cool deal on 11 old Heinlein paperbacks on ebay.
hehehe... I'm turning her into another book-loving-geek in this family of athletes!