The TVTropes Trope Finder is where you can come to ask questions like "Do we have this one?" and "What's the trope about...?" Trying to rediscover a long lost show or other medium but need a little help? Head to Media Finder and try your luck there. Want to propose a new trope? You should be over at the Trope Launch Pad.
This is more about the Character Development than the type of character themselves—reading Arthur Dent, it seems less likely to happen to that character than to Only Sane Man types.
Oh, another example...Midori Days. Seiji thinks his problem is that he doesn't have a girlfriend, but his actual problem is not being able to relate to other people, something he learns as a side effect of having Midori grafted to his arm.
Oh, there will usually be Power of Friendship invoked at some point.
- bump*

Or rather, I've seen it most often in anime...the focus character is a fairly stiff guy, may be The Only Sane Man, who is dragged into wacky hijinks by the other characters against his will, making a mess of his life. But implicitly or explictly, the goal in the story is to make him a better person by putting him through horrible experiences.
May involve a Manic Pixie Dream Girl, but in most of the examples I'm thinking of, it's an entire cast imposing on the stiff guy's time, money and patience.
For a couple of examples, Rozen Maiden and Gash Bell, both featuring boys who hide away from the world until their new companions force them to interact with others.
Explicit in Haunted Junction: the Preacher's Kid who says I Just Want to Be Normal, but is constantly dragged into supernatural mayhem. In the final episode we learn that the school spirits have been tormenting him specifically to avert a Bad Future in which he lives a "normal life" but never emotionally connects with anyone including his future wife and child.
Which seems to lead into a happy ending...except that said preacher's kid was made so miserable by the constant torment that he voluntarily participated in a plan to kill or imprison all the school spirits; and there's no indication that things are going to get better for him.