This is just a sketch of an answer.
It is important to clarify how we define "self-awareness".
The self: There is a huge social psychological literature on the self.
For a scientific review, see for example Ellemers et al (2002).
Even the concept of self-awareness is contentious in that it posits that there is a self to be discovered whereas I assume most researchers would acknowledge that the self is at least partially a social construction.
Psychological well-being: A sense of self-awareness also relates to many other constructs such as having a sense of life purpose or meaning. For more information on that you could look at the psychological well-being literature. At the risk of self-promotion, I can point you to Grant et al (2009) where we reported some of Big 5 correlates of psychological well-being where for example we showed that self-reported purpose in life (i.e., a little bit like self-awareness) correlated well with high extraversion (r=.33), high conscientiousness (r=.45) and low neuroticism (r = -.44).
Self-efficacy: An alternative perspective would be to look at self-efficacy research. Such research often looks at the relationship between a person's predicted and actual performance. Such research often shows that specific performance feedback with practice is important in generating accurate self-beliefs about one's own competence.
Self-other correlations: I imagine also that there is literature that looks at self-other correlations for various characteristics. It would be interesting to see what predicts the size of self-other discrepancies in such domains as personality, and performance. Unfortunately, I don't have any particular references on hand.
- Ellemers, N., Spears, R., & Doosje, B. (2002). Self and Social Identity*. Annual review of psychology, 53(1), 161-186. PDF
- Grant, S., Langan-Fox, J., & Anglim, J. (2009). Big Five Traits as predictors of subjective and psychological well-being. Psychological Reports, 105, 201-231. PDF