Is there a point to Claire Temple?
It should be. She keeps popping up in all the shows, as the only one doing that.
Oconotu
http://acarpenoctem.blogspot.com
She's like Coulson was in the first Phase of the movies, she's a connective thread that indicates these shows are indeed part of the same universe. There's also the added point that she's a version of the Night Nurse, she patches up the wounds of these street level vigilante style superheroes when they need it because they can't exactly go to a hospital.
Nillindeiel
Agent Hill: ...Then aliens invaded New York and were beaten back, by among others, a giant green monster, a costumed hero from the 40's.... and a god.
Agent Ward: I don't think Thor is technically a god.
Hill: Well...you haven't been near his arms.
~Agents of SHIELD; Season 1 Episode 1 "Pilot"
That's certainly one likely explanation, but it also feels like she's being groomed for something.
Oconotu
http://acarpenoctem.blogspot.com
Anything is possible but really she's only being groomed to be the night nurse. She's also not the only character that's been on multiple shows (though she is the most important one really).
Reading my signature constitutes admission that I am correct. (Too late)
It seems she and Rachel McAdams' character in Doctor Strange and likely some third character will be the Night Nurses as part of the underground community from the comics (where Doctor Strange can come in and heal Luke Cage, Nelson & Murdock can represent Peter Parker, Tony Stark can help someone with a bionic arm etc).
Also, this: http://www.imdb.com/board/bd0000130/nest/263626706