The idea of ‘the Impossible Girl’ is because that name in no way applies to her properly. The Doctor thinks she’s impossible and then discovers she’s a perfectly real person, a perfectly ordinary person. His disposition is to treat people like puzzles to be deconstructed and what he learns in dealing with Clara is he was dealing with a real human being and if he’d picked up on that earlier they would have been better friends quicker. So that’s what ‘the Impossible girl’ is, it doesn’t exist, she’s just Clara Oswald.
Steven Moffat, [source]