When Neil Armstrong reported landing in simulators during the training for Apollo 11, he always said "Houston, Eagle. We have landed" or some close variation on that. When he actually got to the moon, he realized that something more poetic was necessary for such a historic moment and ad libbed "Houston, Tranquility Base here. The Eagle has landed" which no one else had ever heard. The communications officer in Houston was clearly surprised but responded "Roger, Tranquility." The name is now recognized by the International Astronomical Union.
Many, many lasting and important inventions were discovered completely by accident. The efficacy of microwaves with regards to heating food was realized when the American engineer Percy Spencer, while working on an active radar set, noticed that a chocolate bar he had on him began to melt.
The Cosmic Microwave Background Radiation, which is the evidential lynchpin in the big bang theory, was discovered unwittingly by the two American astronomers Penzias and Wilson, who did their darndest to get rid of the "background noise" their New Jersey-based radio horn was picking up, going so far as to scrub the entire inside of the horn. Their accidental discovery won them the Nobel Prize, and solidified the foundations of the big bang theory.
Zappo's Potato Chips invented the "Voodoo" flavor by a guy spilling a few spices. Someone stuck their finger in the resulting mess and declared it delicious.
Several of the more iconic moments of George Balanchine's ballet Serenade are examples of this. A dancer falling in rehearsal was added to the ballet, as was another's late arrival.
A lot of extensible, free-as-in-freedom software (some call it Open Source), ends up like this. Prominent examples are Emacs and UNIX (though originally it was proprietary; only with the GNU project has the licensing been cleared up and extensibility preserved). Other examples include the LISP programming language (especially in previous versions, where it was localized in certain areas (e.g. MAClisp (no relation to the Macintosh) in the MIT AI lab) —recently, it's manifested in de-facto standard libraries) , probably because of its built-in extensibility (with defmacro)
The first purpose-built multiplex theater in the US was the result of this trope. When developing a cinema at Ward Parkway Center in Kansas City, Missouri, AMC discovered that the configuration of the center did not allow for one large screen. So they just made two smaller screens instead, and quickly found they could double their revenue by showing two films at once.
The annual holiday practice of "tracking Santa" at NORAD got started because of a typo. In 1955, a Sears advertisement that ran in a Colorado Springs newspaper accidentally printed the phone number for the Continental Air Defense Command in place of a talk-to-Santa hotline for kids. Deluged on his private high-security line by children's Christmas Eve calls for St. Nick, Col. Harry Shoup didn't have the heart to disappoint them, so played along and instructed his staff to give such callers reports on Santa's progress throughout the night. A tradition was born, first as a seasonal call-in line to CORAD and its successor, manned by generations of volunteers from the U.S. armed forces, and currently as the noradtracksanta website.
Theme parks are occasionally subject to something like this. If you ride The Haunted Mansion ride in Disneyland, you may notice a spiderweb with a spider in the Ballroom scene. The Spider is there because, at some point in 1974, some asshole shot a bullet into various attractions, of which only the plexiglass used in The Haunted Mansion could not be repaired properly. The "spider" covers the bullet hole; the result mildly adds to the ambiance (it feels like yet another small detail added to make things spookier).
Much of Don Rickles' materials in his stand up were ad-libbed. Unlike many stand up comedians, Rickles doesn't write his own material beforehand; everything in his act were all made up from the stuff he said or thought up and save them on his noggin for future reference. That's how good he is.
When TVR was developing the Chimera sports car, one of the designers brought his dog to the studio. The dog took a bite out of the front bumper of the clay model while no one was looking. When the designers saw what happened, they thought the notch looked rather good. They smoothed it out and made a corresponding notch on the other side.