COURTNEY WOODS
We were introduced to her as a random student in Clara's class in "Deep Breath" when she offered very wise advice to her teacher. Here we see how we need to watch episodes with a mind to the broad story arc of the season and beyond. As of deep breath, she is simply a random nameless student. However, remember--in the Dr. Who universe, people are rarely given lines, especially important ones, without a reason.
Next she pops up , again as a random nameless student, in "Into the Dalek" with one line as she listens in on Danny's conversation with the school secretary. "She wishes" could be a harbinger line for late in the season.
Finally, in "The Caretaker" she literally bursts in as a full fledged character. I would argue that this entire episode was written to introduce us to Courtney, and give her a name. In one episode, we learn her name, meet her parents, see that she is a "disruptive influence" AND she becomes a de facto compaion with ride in the tardis.
History lesson done (as much as I could do on the fly with my DVR anyhow), Let's discuss where this character is going. I have two possibilities:1-She is a previously unseen regeneration of River Song
There are many clues dangling out there to make me think this.
First...THE HAIR. Simply the hair has been a bit of a running joke towards River
Second...From "Deep Breath", She seems to have a bit of a lingering psychic link with Clara from "The Name of the Doctor"
Third...The last name "Woods"--aka, Forest. River has had many references to Forests. Her first appearance of course included "Forest of the Dead", Her name was translated from an alien language of a forest planet (sorry, no time to Netflix for the specific planet/race). Also, looking ahead in this season we find an episode "In the Forest of the Night"
Fourth... Who is a more disruptive influence in general than River
Fifth...The Doctor seems to immediately trust this little girl for some reason. He is very upset with Danny learning about the tardis, yet he takes Courtney for a ride!!
2-Highly unlikely but possible...She is a regeneration of his granddaughter Susan from season 1
Very few clues here and take some stretching. To be honest, this is based more on my not liking something as big as a missing relative just hanging out there in the 50 year storyline.
First...She is attending the same school
Second..Again, no time to check, but did Ian and Barbara follow here home because she was disruptive? Someone will correct me here i am sure
Third.. Again, the immediate trust the Doctor has in her.
These are my ideas, for better or worse. Feel free to prove me right or wrong, I guess only time will tell....