I'm a big big girl, in a big big world. It's not a big big thing if you leave me. I took it to mean "important" or "influential"... Doesn't this make more sense?

I think the song is quoting and rephrasing the famous (and beautiful) Dylan's song: " You're a big girl now ", at least in in that context the meaning is clear (a woman, not a girl anymore). please have a look here: www.songmeanings.net/songs/view/57336/ I think a big big world is a big place where we can pick whatever we choose, eventually with some struggling. All in all the meaning could be something like: I (now) know who I am, what I want, my life is ahead of me, to be enjoyed. Unless ... the chords chosen by the musician were in a sad minor tone, and other words pointed to a different personal situation. (I don't know the song)

big girl= grown up , not a child anymore

big it means either obese or adult / mature

It's the same as everywhere else. Big means large, 大.