I would also point out there is more to gender then what toys they play with, its been shown that children will gravitate to those gender specific things when given a choice with no input or reinforcement from others.
There is truth to the gravitation statement. A story has been told of parents who made it a specific point to gender-flip their children's Christmas presents that year. They gave their son a doll and their daughter a toy truck. Later that day they walked into the room to see their son scooting the doll across the floor making "vroom" noises, while their daughter held the truck in her arms, rocking it and cooing, "Aw, baby." No less staunch a feminist than Letty Cottin Pogrebin, co-founder of Ms. Magazine, told that story. She talked about it in her book "Growing Up Free," and also mentioned it on a television talk show. But when asked, she would NOT concede even the slightest possibility that there could be something innate in the children that led them to their choices. I wonder how she explains it then, because she never did offer a theory about it.
I question the veracity of that story. She didn't say how old the children were, but I've got a four-year-old grandson and a two-year-old granddaughter, and I can't see it happening with them. It seems to me even very small children will realize that a doll and a truck are two different things with two different purposes. I doubt a child of normal intelligence would ever mistake one for the other. If they wanted to play the specific way they did, I would think most children in that situation would simply trade.
Now, I myself am a strange case. (Which is usual.

) At birth, I was given what I considered my entire childhood to be a boy's name. Although in this country it is actually more common for females, it is solidly a male name in Belgium, Sweden, France, and Norway, and slightly more male than female in Great Britain, Canada, and Australia. I have a much harder time trying to think of female celebrities who have that name, than I do male celebrities. As an adult, I changed my name to one that, it turns out, gets mispronounced more often than not, but the trade-off is that nobody ever, based on my name alone, mistakes me for a man. I like it better that way. There are several men at our church, including our pastor, with names that sound a bit girly to modern ears, and they feel similarly. I have come to just not believe in giving any child an ambiguous name. Don't make them ever have to go through that "Are you a boy or a girl?" questioning, that I found so embarrassing.
It wasn't only my name. My parents also made me keep my hair short when I wanted it long, and wear jeans and t-shirts when I would have chosen dresses. They pushed me to get out and run, climb trees, and ride bicycles when I preferred to play with dolls and domestic toys. As I got older, it was a struggle to be allowed to do such things as shave my legs and underarms, tweeze out my obvious mustache and unibrow, and wear makeup, jewelry, cologne, and nail polish. One might think my parents actually wanted a boy instead of a girl, but I had two brothers, so I couldn't figure out why they were trying to make another boy out of me. My best guess is that, since my upbringing encompassed the late 1960's to the early 1980's, it was the thick of that "Ms." era, and my parents decided I would be "equal" to my brothers whether I wanted to be or not! Yes, here again I have to point out that "equal" does not mean "identical."
This said, if a girl wants to keep her hair short, wear jeans, play with toy trucks, and call herself Bobbie if her name is Barbara, I can't think of a good reason she shouldn't be allowed to.