Are rebel figures in children's literature important because they destabilize some of the existing cultural values or because they promote consciousness and new awareness?
Few things in life--or literature--are strictly black or white, and that is what your question suggests: that rebel characters are important because they either "stir up" cultural values or create greater awareness. The answer must be both, it seems to me, because doing one of them will necessarily accomplish the other.
One of the greatest examples of this can be found in Little Women by Louisa May Alcott. The character of Jo March is the ultimate rebel, particularly when she is contrasted with her three sisters and the expected role of women in the Victorian era. The girls' rather modern-thinking mother wants this for all of her daughters:
My dear girls, I am ambitious for you, but not to have you make a dash in the world—marry rich men merely because they are rich, or have splendid houses, which are not homes because love is wanting.... [B]etter be happy old maids than unhappy wives, or unmaidenly girls, running about to have husbands.... Leave these things to time; make this home happy, so that you may be fit for homes of your own, if they are offered you, and contented here if they are not.
All of the girls assert their independence in small ways, but it is Jo who becomes a rebel by living out these things and more.
Jo is a writer. She is the playwright and director of plays for her siblings; while they will all play the male roles, it is Jo, who revels in the swashbuckling, mustachioed, cape-wearing male characters in her plays. She impetuously chops off her hair, selling it to pay for her mother's trip. Later, when she goes to New York to begin her literary career, she writes very non-literary thriller stories for a rather trashy magazine because they sell. She writes them anonymously both because she is a woman and because they do not reflect her moral values. This is a compromise that she makes at first but soon realizes (with the help of Professor Bhaer) that she is not writing in her own voice. Bhaer also invites her to a scholarly discussion in which she is encouraged to express herself; her ideas are well accepted. In fact, they tell her she should have been a lawyer, and she says, “I should have been a great many things.” Jo turns down the love of a rich boy because she does not love him as a wife should love her husband and is prepared to live her life alone; eventually she falls in love, but with one of the poorest men she could have found (though Bhaer certainly has many other redeeming qualities).
All of these things are demonstrations of Jo's breaking the stereotypes of women in this time. She is not docile and subservient; instead she is hot-headed and impetuous, even cutting off her hair, the crowning glory of a woman's beauty. She is not content to settle for a man she does not love in a romantic way simply because he is rich, and she pursues her writing career, a profession dominated by men.
Jo "destabilize[s] some of the existing cultural values" but also "promote[s] consciousness and new awareness." In chapter one, Jo says:
I hate to think I've got to grow up, and be Miss March, and wear long gowns, and look as prim as a China aster! It's bad enough to be a girl, anyway, when I like boys' games and work and manners! I can't get over my disappointment in not being a boy; and it's worse than ever now, for I'm dying to go and fight with Papa, and I can only stay at home and knit, like a poky old woman!
She is a rebel not because she wants to be a boy but because she wants to be free to do as she pleases.