7. River Song
I didn't think anyone would ever beat Donna Noble as my personal favorite Doctor companion. Then I watched "Silence in the Library". I heard those famous first words, "Hello Sweetie", and I was in love. From the perspective of a writer, the capacity to establish (believably) a character who has infinite history with the main character, but has never appeared before, and then bring their story arc to a tragic, cathartic, wondrous conclusion in the span of a single two-part arc, is just plain amazing. I have watched, and rewatched that two-parter trying to figure out just how they accomplished it! In one fell swoop, they created a legend to rival that of the Doctor's, honored all that had come before, and made us giddy for the future of the series. Why is this a problem? In a nutshell, Season 6. Sure, we had Matt Smith's Eleven before that, who never seems more wooden than when he is next to the subtlety and charm of Alex Kingston's River Song. And that was bad. But then came Season 6. Spoilers below. Season 6 is when we find out River Song is Melody Pond. And that she was brainwashed. And that she lost 30 years of time with no explanation. And that she grew up with her parents as her best friends but never showed any care about them in the show at all until we found that out. And that she kills the Doctor, by being put in an astronaut suit for no reason, by forgetty aliens and their one-eyed boss lady. And that she instantly breaks free of her assassin programming (when she kills him the second-but-actually-first-time) just by the Doctor suggesting she might one day sort of like him. And, and, and. The list of objections goes on, and have been enumerated by angry fans time and time again. The ultimate problem is that the character feels like she is written on an episode by episode basis, when she shouldn't be. The producers, writers, and even actors have known that River Song would have to be an integral part of the canon since her debut. Why wasn't she planned for? Why does she not make sense? If we, the viewers, can poke so many holes in her story from our armchairs, why didn't the editors? Why didn't the writers? So how can we fix this? This is the million dollar question. And if it can be answered well, and handled correctly, it would restore our faith in the storytellers, particularly in series helmsman Steve Moffat. Because that is the ultimate cost of flubbing something as vital as River Song: Fans don't trust your capabilities anymore. We are less likely to buy in to your new arcs, and we are wary when you announce you are laying your hands on an old one. In order to fix this, creators will likely need to rely upon the "wibbly-wobbly" nature of the Whoverse to implement any satisfying solution. The problems in continuity and character development are so pervasive that a massive rework is necessary. However, one of the great strengths of the Whoverse as a canvas for storytelling is that those major reworks are completely possible, with the appropriate explanation. Saving River Song from her the tragic festering Melody Pond infection would be a Christmas-special miracle. I would even go so far as to venture that almost any radical explanation that erases her current revealed arc would be well received by fans. Reboot her timeline. Or maybe she isn't really Amy's daughter, maybe she's an entity in the Tardis that manifested through Amy? Anything is possible. Then give her more screen time. Clara's great, but shouldn't River have been this season's primary companion? Give us more of River's history to justify what seems now to be her sick delusional attachment to a man who WAS very good when he was Tenth and before, but who now routinely lies to, betrays, and otherwise unnecessarily harms those he loves in order to save himself discomfort(read: letting River languish in prison for no reason). And if all else fails, ask the fans. Hold a poll, or read some good fan-fiction, or otherwise cave and give us a resolution which heals the festering infection of Melody Pond which River Song has been slowly dying of since Season 6 began.
http://youtu.be/pYQAv_-txhM