[Video at bottom. Girls dancing fully clothed, but semi-provocatively, so may not be SFW.]
Something just occurred to me -- the reason that females are so emotionally unstable during adolescence is likely a by-product of selection for increased exhibitionism during the years that they are in their physical prime and are in search of the best mate, long before they must learn to settle. Exhibitionism isn't a basic personality trait, but rather the result of having both high Extraversion and high Neuroticism. The former means how outgoing and socially engaging you are, while the latter means how emotionally balanced vs. volatile you are. A high-N person experiences more anxiety and insecurity than a low-N person, and it doesn't take much to get them worked up over something. A high-E person directs most of their energy outward to the social world, rather than prefer the company of themselves. So, someone who's emotionally unstable and insecure, but who also feels compelled to seek out social stimulation and to direct their energy outward, will be something of an attention-whore. "Hey you all -- validate me!" Also, because they are easily worked up, yet spend lots of time around others, they'll come off as a hothead as they respond to all the little things others do that don't please them.
Assuming girls of a certain age are in competition with one another for the attention of the males who matter, then natural selection favors those individuals who can best upstage the others. To do this in an environment where the alpha-males are primarily interested in looks and promiscuous sex, she needs an exhibitionistic disposition, which requires both higher E and N. These are the main changes in average female personality during adolescence -- girls become more outgoing than boys, and more emotionally unstable as well. The evolutionarily puzzling nature of the second phenomenon makes sense once viewed in interaction with the first -- this combination of traits gives the girl a leg up in the struggle to ensare a guy's desire. Given that guys find pretty much any girl worth mating with, it takes an extra effort to direct his attention her way and keep it there. (In contrast, the average girl isn't simultaneously thinking "I'd mate with him" about a wide variety of guys, so males aren't under the same degree of pressure to show off so nakedly. This might not be true where girls do judge guys heavily on looks, like pathogen-ridden areas.)
If this is correct, it would represent something similar to what Harpending & Cochran (2006) call "adapted genotypes," where the fitness boost comes with a particular genotype (here, it would be alleles favoring high-E and others favoring high-N). They are the first responses to abrupt changes in selection pressures, thus also bringing ugly costs that are nonetheless outweighed by the benefits (sickle-cell anemia in malaria-stricken areas, for example). These are distinct from "adapted genes" where a particular allele provides benefit and little or no cost -- since they're so optimal, it will take longer for selection to find them. If there was a sudden, recent shift in Western societies that selected for exhibitionism, the sloppy first solution might be to increase the frequencies of alleles conferring higher E and N, even if non-exhibitionist-related aspects of high-N otherwise brought costs (like greater likelihood of experiencing depression).
As far as I know, though, this change is pretty recent -- like within the past 100 years or so (Victorian girls were prudish) -- so even the ugly first genetic solution probably couldn't have been hit on that quickly. Rather, the sudden increase in high-N females could be due to facultative strategies that the female pursues to cope with her environment. That is, she figures out that the mating system is a serial monogamy / de facto polygyny, realizes by instinct what this means for which traits a guy will focus on in a girl, and then adjusts her level of N upward accordingly. This would constitute part of the environmental contribution to complex traits like personality. Over time, assuming the pressure remained, selection would favor females who were genetically predisposed to high-N rather than those who'd have to figure it out on-the-fly, but in the short-term this facultative approach might be more common.
And now to the fruit of a little fieldwork on YouTube to document this phenomenon. For the sake of decency (....), I'll only put up one clip. The girl on the right is such a cocky little hothead, and just look at how greatly that enhances her dance moves! (Also note the blatantly competitive nature of their dancing.) The user's name has the words "mexirican," so they must have fiery Latin temperaments. I realize the music is Idiocracy-level trash, but not all music is meant to excite the intellect. I also anticipate the objection that "If they were your daughters, you wouldn't want them behaving like that!" Half-true. I would mind it if it led to promiscuity -- but if they were just teasing, then that would be fine.