Just my personal experience: My mother was given this option when I was in kindergarten at age 4, to skip me ahead a grade, and she said no because she was worried about me fitting in with older kids, making friends, etc. 
It turned out that I was small for my age for the next ten years anyway, and always seemed to be a head shorter and 20 lbs. thinner than anyone my own age. And I still got picked on, not for being young/smallest, but for being a smart-aleck and a geek! 
What really helped me prepare for college and make friends was going to an all-girls boarding school. The boarding school was run a bit like college but with more supervision obviously, and had a more flexible class scheduling option so you could take extra instruction if you were really good at a particular subject. You weren't allowed to fill up your schedule with study halls like the public school kids did. Since it was single sex, there weren't all the silly social games and flirting nonsense going on in class--classes were for learning, flirting was to be done under the headmistress' supervision at dances with approved boys' schools. 
A few years later, when a couple of teachers were nevertheless fed up with my smart-aleck ways, the school guidance counselor recommended I take part-time classes at the local university. Upon reviewing my PSATs, SATs and transcript, the university admissions office told my mother that while it was too late to apply for financial aid for that particular semester, I could be accepted and given a hefty scholarship if I wanted to start as a full time student there in the spring. I was 16, and my mother felt that I should have a high school diploma rather than skipping years of high school to go straight to college, so I had to suffer through two looooooooong horrible years full of teasing and bullying that were of absolutely no benefit to me in any way whatsoever. I did not make long-time friends, have unforgettable experiences, or learn bugger-all. I was bored stupid, to the point that my grades went in the toilet out of sheer contrariness and boredom. When I finally got that stupid diploma, I no longer qualified for the scholarship I was originally offered! So had to work for a few years in order to afford to go to college, and worked my butt off at part-time jobs all through college, too--I could've done WAY better if I had had time to study instead of working 35 hours/week. 
Now I look back on it, and wow, all the things I could have done...I would've been done with my undergrad at age 19, gone straight to grad school...probably ruling the world by now...
Seriously, do not hold your daughter back. Let her be awesome. Maybe someday, she'll be president.