Kosa mi je postala bela ---- is correct However, we prefer to say "Osedeo sam" or "moja koda je postala seda" İf you say "Kosa mi je postala bela " we could think it is because there is too much snow on your hair or flour or anything other with the same colour. Apart of this Nina has already given you a good explanation.

It determines the subject, which is "me" or "my". If you say "Kosa je postala bijela.", others would ask "Čija (kosa)?" (Whose hair?). This sentence is a variation of this form: Moja kosa je postala bijela. "mi" = "moja" in this case If you say it without it, it could translate to "Hair has turned to white." I hope this helps!