You Will Be Beethoven

You Time Travel into the past; something happens as a result of your time travel, and somebody dies. Somebody important and probably even famous, who you know stayed alive until after this date. What can you do to close the Stable Time Loop, Set Right What Once Went Wrong, or at least trick out time so you can end up back in a "Close Enough" Timeline to your own? Why, impersonate them, of course!

Usually, for added drama, the person you're replacing is supposed to die in some other way in order to trigger a significant event, meaning that you expect replacing them to lead to a Heroic Sacrifice. This is not to say that it actually does, though; you can often find a third option besides dying nobly in the past and screwing up the timeline.

There is also a variation in which the time traveller doesn't cause the historical figure's death, but instead discovers that the historical figure never actually existed; the time traveller then has to impersonate the historical figure in order to create the history he remembers.

Compare and contrast Time Travel Escape; in particular, the third option in the more-dramatic version often involves pulling something like a Time Travel Escape. Also compare Beethoven Was an Alien Spy, which is a supertrope to at least the Stable Time Loop version of this, and Emergency Impersonation, a very similar plot that doesn't require time travel.

Comic Books

 * Applies to the (Pre-Crisis version of) DC Comics' heroine Superwoman. She was a time traveling researcher who came to the present from the 29th century in order to try to find Superwoman's secret identity, the only superhero whose secrets were never discovered. She ends up realizing that she was supposed to be her. At first she only visited the past when recorded history said Superwoman appeared, but a time travel accident left her stuck there with amnesia for years. She eventually returned to her home time, however.
 * This sort of happens in issue #5 of the original run on Fantastic Four, in which Dr. Doom (his first appearance) sends the foursome back in time and The Thing, who has always been a fan of stories of Blackbeard, ends up becoming Blackbeard.

Literature

 * In Night Watch, Vimes spends almost the entire novel impersonating his own mentor John Keel.
 * Happens several times in the Time Wars series:
 * In The Pimpernel Plot, a time traveller's action causes the death of Sir Percy Blakeney, a.k.a. the Scarlet Pimpernel. One of the Time Commandos is assigned to replace Sir Percy to ensure that the events described in The Scarlet Pimpernel play out as they should. At the end of the mission, another agent is assigned to take over and live out the rest of Sir Percy's life.
 * In The Zenda Vendetta, the villains murder Rudolf Rassendyll, again obliging one of the Time Commandos to take his place.
 * In The Ivanhoe Gambit, Sir Wilfred of Ivanhoe isn't dead -- the heroes deliberately kidnap him so that one of their number can impersonate him as cover -- but the timing means that the impersonator does end up having to do all the stuff that Ivanhoe is remembered for (meanwhile, in a subplot, a supporting character becomes King Richard the Lion Heart after the real Richard is killed by the villain).
 * Perhaps because this imposes a restriction on the plotting, the other Time Wars novels just have the heroes befriending historical figures, not replacing them.
 * In Isaac Asimov's The End of Eternity, someone is sent back in time by the leaders of Eternity, a centuries-spanning time-traveling organization, to explain the workings of time travel to its purported inventor, thus completing a Stable Time Loop. The inventor dies, however, and the person from the future ends up replacing him and "inventing" time travel himself. Or rather, this was the plan of the leaders of Eternity, based on secret information about what had happened in the previous iteration of the loop, but in the end of the book.
 * In Tim Powers' The Anubis Gates, the main character, expert in the study of a (fictional) Victorian poet, travels back in time and ends up being said poet in the past. This is the variant version mentioned in the description, as the poet in question didn't die and need replacing. He'd always been the time traveller, product of a Stable Time Loop with no discernible origin. Nobody ever composed his poems, either, which worries the protagonist more than the actual replacement does.
 * Behold the Man by Michael Moorcock. You can probably figure out the subject in question from the title.
 * In one of Poul Anderson's Time Patrol short stories, a time-traveler gets pressed into taking the place of an assassinated Persian royalty; history remembers him as Cyrus the Great.
 * Another story in the same series, which has the magnificently awesome title "The Sorrow of Odin the Goth", has a time-travelling anthropologist trying to find the source of a particular legend involving the god Odin. He visits a dark-ages Goth community several times over the course of decades, and the locals, noting that he never seems to age (among other reasons), decide he is Odin. At the end he has to close the Stable Time Loop by doing what Odin is described as doing in the legend, even though it means killing two of his grandsons.
 * The variety where the famous person never existed appears in at least two writings of Stanislaw Lem.
 * The protagonist of Manly Wade Wellman's Twice in Time travels back to Renaissance Florence to meet Leonardo da Vinci and to leave a mark on the back of the Mona Lisa to prove he made the trip. He eventually becomes Leonardo da Vinci and paints the Mona Lisa itself as proof of his trip. Of course, the time traveler's name was Leonard to begin with.
 * In the Found series by Margaret Peterson Haddix, this is combined with Time Travel Escape. A group of children who are adopted, but cannot find their birth parents find out they are long lost children of the past (such as Anastasia or Virginia Dare of the Roanoake Colony) that were taken by time travelers from the distant future. Unfortunately, the time machine went wrong and ended up at an airport in the 90s. In the second book, Sent, this trope is almost played straight when Chip and Alex are found to be missing medieval princes who are meant to be killed.
 * In The Door Into Summer by Robert A. Heinlein, a professor who's invented a faulty form of time travel talks about how one of his students took the risk and went back in time. His name? Leonard Vincent. While he never appears in the story, so we never find out if he really is da Vinci, the protagonist wonders if Leonard, trapped in the past, drew plans for flying machines and such because he was hopelessly trying to recreate things he'd seen in the 20th century. He also quietly mourns for the man, talking about how incredibly impressively difficult it would have been for him to travel across fifteenth century, pre-Columbian America and wrangle his way to Europe, and pitying him as a Fish Out of Temporal Water either way.
 * In Harry Potter and the Prisoner of Azkaban, Harry thinks he sees his father send a Patronus across the lake to save himself and Hermione. When they go back with the Time-Turner, Harry goes across the lake, hoping for a glimpse of his long-dead father...and realizes that he was the one who conjured the Patronus.

Live Action TV

 * Star Trek:
 * In the Star Trek: Deep Space Nine two-parter "Past Tense", Sisko ends up in the 21st century as the result of a transporter accident, where he has to impersonate the soon-to-be martyr Gabriel Bell after accidentally causing his death in a food line. At the end of the episode Starfleet Command has some questions to ask about why the picture of Gabriel Bell in the history books resembles him so closely; it gets Lampshaded again in a later episode when Nog is looking at a historical database and comments that Gabriel Bell looks an awful lot like Sisko.
 * In an early draft of the Star Trek: The Next Generation episode "Yesterday's Enterprise", Surak was accidentally killed by time-travelers, and Sarek had to take his place in order to bring logic to Vulcan. The actual episode is a borderline example. Tasha ends up going back in time with the Enterprise-C after she dies, but it's not to fool anyone into thinking she's Captain Garrett -- it's just that they need somebody to captain the ship.
 * Deep Space Nine also has a strange example with a short timeframe. In "Visionary", O'Brien is time-jumping uncontrollably, and he sees the station blow up in the future. To prevent this, he decides to force a time-jump, find out how the disaster happens, and come back. Unfortunately, the device that makes this possible emits deadly radiation, and O'Brien overestimates his endurance; when it's time to go back, he realizes he won't survive the return trip. Solution? He gives his future self the device and sends him back. The disaster is averted, and at the end of the day there's still just one O'Brien, but he can't help wondering if he's really the same man.
 * In The Twilight Zone (the 80s version), the episode "Profile in Silver" has this as a plot. An academic from the future travels back in time and ends up saving JFK from the assassin's bullet. As a result, this creates an alternate timeline where the extinction of the human race via nuclear war is inevitable. Kennedy is informed of this and offers to go back to be assassinated in order to restore the timeline, only for the academic to send him to the future instead while he takes Kennedy's place in the limo and dies. We then see Kennedy in the future, giving a university lecture praising the sacrifice of heroes like the professor.
 * Another episode of the '80s The Twilight Zone qualifies as well. In "The Once and Future King", an Elvis impersonator goes back in time and meets the younger Elvis, who mistakes him for his brother. The impersonator is frustrated when Elvis shows no interest in becoming a rock singer, which eventually lead to a struggle that leads to Elvis' death. The impersonator ends up taking Elvis' place.
 * In Red Dwarf, the crew go back in time and accidentally kill Lee Harvey Oswald, preventing the Kennedy assassination. After the usual time travel shenanigans, eventually they convince a JFK from later in the time line to take the place of the assassin, firing from the grassy knoll.
 * In Heroes, Hiro goes to 17th century Japan to find that his childhood idol, Takezo Kensei, doesn't even come close to meeting his expectations. While not actually doing (some of) the amazing things that Kensei would be credited for, Hiro definitely plays a part in the making of history. In fact, the writers actually try and fake out the audience to think this is what Hiro is going to do after . However.
 * There's a TV-movie (read: Poorly-Disguised Pilot) that involves a NY cop being hunted by time-travellers from the past, to prevent him from realising his destiny -- to go back in time to become... wait for it... Nostradamus. Unlike most examples, the Stable Time Loop is averted, except for a woman with Ripple-Effect-Proof Memory.
 * A key plot point in Babylon 5 involves Sinclair going back in time one thousand years to intervene in the previous war against the Shadows.
 * Farscape: Aeryn tries this in "Different Destinations" to save the life of a bumbling cook who clearly had no place in a battle. He doesn't let her, though, and dies like he's "supposed" to. Still doesn't fix the timeline, though...
 * In Doctor Who, Vicki decides to leave the TARDIS when she falls in love with Troilus, becoming the Cressida of Greek mythology.

Tabletop Games

 * The Thespians in Continuum have this as a big part of their schtick. The subject doesn't necessarily have to be dead, either, just unavailable for their historic moment.

Video Games

 * In Gamelet, a currently-in-development video game, you play a time traveler who ends up having to act out the plot of Hamlet, impersonating the title character.
 * Jigsaw forces you to be the instigator of some of the twentieth century's worst disasters, starting with assassinating Archduke Franz Ferdinand.

Western Animation

 * In Futurama, Fry eventually becomes his own grandfather after accidentally killing the person who was supposed to be/he thought was his grandfather.
 * In a recent Family Guy episode, Stewie discovers Leonardo da Vinci was his ancestor, and his archnemesis travels back in time to kill him, therefore eliminating Stewie's existance. In the end, Leo dies, so Stewie must continue living in the 16th Century as him until he can pass his progeny forward, after that, he built a cryogeny chamber and froze himself until the present day.

...and I'll be Bach.