Relationship Reboot

Two characters who since they first met have been at odds with each other—or even active enemies—decide in the wake of some event or epiphany to wipe the slate clean and start their relationship anew. To symbolize this new beginning, they re-enact a new "first meeting" between themselves by shaking hands and trading introductions. Sometimes they also trade capsule descriptions of themselves.

Not to be confused with Relationship Reset Button.

Examples of Relationship Reboot include:

Comic Books

After the Infinite Crisis, Batman returned to Gotham City and decided to give the former corrupt cop Harvey Bullock another chance.

Fan Works

Near the end of the Ranma ½ fanfic Equal Halves by Deborah Goldsmith, Ranma and Akane, their lives and attitudes permanently altered by their brief encounter with Ranma's female counterpart from another universe, re-enact their first meeting after Ranma muses their relationship would have proceeded much better had he met her when he was in his male body.

Film

Blast Hardcheese and Doctor Lady do this in Space Mutiny right after she shows off her, uh, dancing.
Invoked in Mr. and Mrs. Smith. Both John and Jane find out that the other is a paid assassin and, by extension, that everything they've told each other about themselves is a lie. Each starts telling the other the truth about their real selves, and John notes, "We're going to have to redo every conversation we've ever had."

Literature

In Stardust by Neil Gaiman, Tristran Thorn does this with the star.

Live Action TV

In Scrubs, the Janitor has offered this to JD a few times. It's almost always a trick, and even if it isn't, Failure Is the Only Option for all their attempts at a relationship reboot.
On Happy Days, Richie and Lori Beth do this, complete with reenacting their original Meet Cute dialog.
Pandora begs Thomas to let them do this (after her cheating with Cook) at the end of Katie And Emily's Skins episode.

Pandora: Please, can we start again? I promise, I promise I'll never...
Thomas: Hello, I'm Thomas. So glad to meet you.

Theater

In The Most Happy Fella, Tony and Rosabella go through "Happy To Meet Your Acquaintance" in their first scene together since their wedding (which took place just after Tony was badly injured in a truck accident). Even more poignant is the final scene, where Tony recalls (in words and music) the note from him that Rosabella (or, to use her real name, Amy) read off a restaurant menu in the first scene.

Web Comics

Towards the end of the Sister arc in El Goonish Shive, Ellen and Nanase do this. This is an unusual example, because Ellen is a magical Opposite Gender Clone of one of the main characters, and despite having all of Elliot's memories, Ellen is an entirely new and different person without a friend in the world, which is causing her quite a bit of angst. Slightly subverted, as Nanase doesn't get it at first. Ellen then goes on to reintroduce herself to the entire cast, with Nanase threatening to bitch slap anyone who laughs.
At the end of the fifth chapter of Cat Legend, Sarah does this with Dorian. The congeniality of their second introduction is especially ironic in light of the fact that they are both sporting highly visible facial injuries they inflicted on each other.
Something*Positive, Halloween 2004.

"Hello, I'm Peejee."
"Hi, I'm Mike. Sometimes I'm an asshole."
"That's fine by me. The interesting people always are."

In a very literal case, at various points in Diesel Sweeties, the main character has literally rebooted her (robot) boyfriend after they'd broken up.

Western Animation

Played with in an episode of Family Guy, where Quagmire lets slip one of his fantasies about Lois to Peter:

Quagmire: Is it possible she's a whore? ...You know, like on weekends, just to pay for her mom's dialysis? ...Like in my fantasy? ...Y'know what, let's start over. Hi, I'm Quagmire! (offers his hand)