Most of the stuff I'm thinking of is non-gendered. So I'll hit those few exceptions first and then go on to generic kid stuff. Biggest average differences between boys and girls here: I think physical maturation rates make a big difference when you're smart. It's very, very different in many social situations to be a girl who is noticeably younger than her intellectual peer group than to be a boy who is younger than his. When I was 15 and in a math class with 17-to-19-year-olds, I was not visibly younger than they were. I was not on the average smaller. Etc. Boys my age would have been. A lot of the college friends I've stayed in touch with were two class years ahead of me, which makes them mostly three calendar years ahead of me. Like markgritter. And look: sure, it's quite possible that if markgritter's and my ages had been reversed, we would still have ended up married. But it's a lot less common for teenagers to have figured out that boy older/girl younger is more common rather than always better. Up side from girl's point of view: romantic involvement with older peer group much easier as a younger girl than as a younger boy. Down side from parent's point of view: same. (On the flip side, it might be a relief for a 15-year-old to just do their math and get treated like a kid sibling. I don't know. I got some of that, mostly from girls, and sometimes it was really nice.)
The other thing I'm thinking of off the top of my head that's gendered is the form social limitations take: mostly girls are told they can't do things because girls can't do those things, where boys are told that boys don't do various things. I'm not convinced that one is easier to deal with than the other. But they're different, for sure. As the parent of a smart girl, you will almost certainly hear some little rat of a boy who is not nearly as wonderful as your wonderful daughter tell her that she can't do something because she's a girrrrrrl. Resist the urge to dash his brains out against the nearest wall. Let your daughter do the dashing out of brains; she will need the practice. (Also, it may be that I am just so overwhelmingly femmey that it was too ridiculous to even float as a trial balloon, but nobody ever suggested that it meant I wasn't a "real" girl when I did something girls "can't" do. Boys who do things boys "don't" do are not generally so lucky. It's a lot of hard work going around and doing the things you "can't" do to show that you can, but sometimes there is an outcome that makes sense.)
The rest of it...I keep thinking how my ideas of what worked well for me as a smart kid aren't all that different from my ideas of what works well for a slower kid, or a special needs kid, or a special needs kid who is also smart, or an all-around-average. Maybe I just don't know because I was always someone who was recognized by peers as smart--it didn't surprise me that this person had categorized me that way, because, well. And that's how my parents were: it was not, "Oooooh, we have this precious wonderful gift we must nurture!" I mean, they did nurture my gifts. But they didn't treat it as all one thing--rather than being smart, I was someone who was quick with numbers or picked up languages easily or whatever was applicable at the time--and they didn't treat it as in any way surprising or fragile. Okay, smart kid. Okay, kid with brown eyes. Okay, kid with a pleasant-but-not-outstanding singing voice. They were just matter-of-fact about it.
And observant. I think one of the things my parents did for me--and a very few of the best teachers did, too, but not many--was to observe what I was capable of rather than assuming. They gave me lots of opportunities to talk things over with them and to pick my own reading material and my own activities. They watched what happened. They went from there. There was never a point where they sat down and decided, "Okay, an average kid can read this book at age x, and our kid is smart, so she will be able to read it at x-2." They let me pick books. They talked to me about books. They talked to me about other stuff. They listened when I talked. We went from there.
In junior high I had been in a special math class for kids who were ahead of the normal "accelerated" math class. It was maybe a dozen of us, maybe a few more than that? Anyway, we were all quick learners and it went great, and I loved it. I think we all did. In high school, we were mixed back in with the rest of the school at the level we were taking--so we were in Trig, and so were the "normal smart" kids the year ahead of us, and so were the "average" kids the year ahead of that, and so were a few of the kids who were slower at math but wanted trig for college prep their senior year. That was a substantial mix of abilities, and I was bored out of my tree. I told my dad. He remembered a math prof he'd gotten along really well with in college, and he contacted him and asked: what's the biggest gap you get in smart kids who come in having taken calculus in high school? What do you wish they knew? And the guy said, funny you should mention that, we don't feel like a lot of kids are coming in having learned a lot of mathematical thinking, they're just doing calculations. So we put together these supplemental packets for freshmen who are through calc but not really where we want them to be. Here's one for your kid.
So what the folks did right in this example: they listened. They treated me like a person throughout my childhood so that I knew they would listen when I had something to say and not dismiss "this class is way too slow" as "I don't feel like doing my homework." When they weren't sure what the next step should be, they asked for input from someone else. They trusted their instincts about whether that input was relevant. And to the best of my knowledge they never once said, "Is it okay that this thing is designed for college freshmen and our daughter is 14?" It was relevant to my interests, it was available, it was not in a form that was going to distress or pressure me if it didn't go well. That was what was relevant, not my age.
I don't think I see very much of how that would have been different if there was a supplemental math packet that was what I needed at 14 that was designed for 10-year-olds instead of for 18-year-olds. You know? My experience of the world was certainly different than a person's in that situation would have been, but the steps my parents were taking didn't have to be. They didn't make the situation, "You are so smaaaart, you are so much smarter than any of your classmates that you need this and they don't, because boy howdy are you ever smart." It was just a thing they could do that would be good for the specific kid they had, so they did it.
So, o asker of this question, maybe it'll be like that with your little girl. Maybe it won't. Maybe she'll start asking questions about botany, or about Chilean history. Maybe she'll get enthralled with revenge tragedies. Maybe she will think very hard about how her soccer practices are organized and figure out ways that could and should go better. Maybe she will start begging you for violin lessons at a very young age and not give up until you do. Maybe she will spend hours IMing her friends about the minutiae of an anime series. Kids come up with their own stuff if you let them try stuff and pay attention to what works for them. I loved my chemistry set and didn't do much with my telescope. Don't get too personally invested in any one thing working for this particular kid you've got. Hang in there and enjoy the time with her.
I'm not sure this was actually decisive, and it's certainly too late for the person who asked the question to do this with their child, as she is too old, but when I was a newborn baby, my dad would pick me up and whisper to me, "Science is fun. Math is easy." Did that make a difference against pervasive social messages that science is dour and grim and math is impossible? Possibly not, but having a dad around who thought that was the natural thing to do sure did.
My mom once told me, "We knew you'd never be 'normal.' What we could give you was that weird was a good thing at our house." And they didn't do that by making me feel like a precious hothouse flower, they did that by pointing out the cool weirdness in all sorts of people and things. By appreciating human variation, really appreciating human variation rather than deciding that the narrow vision that attaches to the supposed average of humanity is bad but some other narrow vision is good.
*While I am not offended by lj messaging or Facebook messaging, I strongly prefer e-mail.