I hate to consult the tv for answers, but here is tv tropes website definition for stories about "wanting to be normal". If you're helping your daughter develop her charachter then you need to explain Greek pathos to her and tell her to be truly pathetic (in a good way, so everybody cares) she needs to actively dig for her epiphany and not just let the phrase become cheap.
http://inpraiseofargument.squarespace.com/teach-a-kid-to-argue/


"i want to be normal"
When done well, this can be an interesting metaphorical exploration of how even the most blessed can feel isolated and abnormal. When done poorly, it comes off as cheap angst that will get tiresome.
http://tvtropes.org/pmwiki/pmwiki.php/Main/IJustWantToBeNormal

Well, there's my weird2 cents. Why can't I just give a normal answer for once. It's like I'm always talking in metaphors where it almost sounds like I know what I'm talking about, but I can never be sure.

Here's a recent forum discussion (from here) which equates this phrase as loosely related with grappling with existential depression, possibly leading in some cases to Dabrowski's positive disentigration:
http://giftedissues.davidsongifted.org/BB/ubbthreads.php/topics/152956/1.html

Youth lives by personality, age lives by calculation. -- Aristotle on a calendar