3. She Is Humble And Curious
Companions to the Doctor have always had certain high-levels of curiosity, almost a prerequisite if you ask any viewer. But what sets Clara's apart is that she tries to keep it dialled. We can tell she is very curious, yet she keeps a control over it so as to not get overwhelmed and maybe as a defense mechanism. She acts as if she doesn't deserve it and says nothing of it to validate herself. We all know that the queen of I'm-special-so-the-Doctor-chose-me was Rose (sorry Rose fans), Clara doesn't show just how unbelievably wondrous her situation is and how excited she is about it (which takes a lot of strength), afraid it will go away. She is acutely aware of this chance to travel the Universe and she isn't (or hasn't yet started) taking it for granted. She is humble enough to be unassuming and curious enough to ask.
2. She Finds The Doctor
After recent episodes, we've been shown that the Doctor just can't find Clara whenever he wants to, she finds him. This might, of course, be a Moffatt Impossible Girl thing which no doubt will be made clear as we discover the Whoverse through Moffatt's puppeteering. But it's a good change of pace that the Doctor just can't find this mysterious newcomer as he wills, and that she just happens to run into him all the time (so far) with good timing. You gotta love this fresh take on the basic "companion", she goes to him (willfully or through a fate of her own) if and only when he stops looking for her and stays put. It just adds to the mystery of it all.