After traveling back in time, it's accepted protocol that, instead of giving out your real name, you should use an alias that refers to a future aspect of popular culture. It's just how these things are done. If you really went back in time, this probably wouldn't be the brightest thing. After all, what happens if someone remembers that name or writes it down somewhere? In Movie Land, however, this is justified by the Rule of Funny.
When there's no Time Travel involved, and using such a name would be a lot more risky, it's Character Name Alias.
This becomes an important plot point in the Vertigo comicBrave Old World. After the characters realize they're not the only people transported back to the year 1900, they round up the dozens of other refugees by announcing a talk to be given by Mr. John Lennon and Mr. Mick Jagger.
In the fifth episode, First Citizen Brown tells Edna that Marty's real name is Yakov Smirnoff, and when Edna is forced into the past, she assumes the alias Mary Pickford.
Inverted and subverted by Time After Time, where H.G. Wells comes to the present  and assumes the name "Sherlock Holmes", mistakenly thinking that the fictional character would be forgotten in the future.
Plus, due to not many people knowing what "H.G." stands for, he's nicely able to fit in just by using his real name, Herbert Wells.
Inverted in the first Austin Powers film. Having missed out on thirty years of pop culture, Austin picks the aliases Richie Cunningham and Oprah for himself and Vanessa. Similarly, Dr. Evil, having gone through the same freezing process, calls his moon laser the Death Star and the task of assembling it The Alan Parsons Project. Having gone back to 1969, however, only Scott realizes the idiocy.
Dr. Evil also uses footage from the film Independence Day to frighten the president and others into thinking the White House is being attacked. But tells him that's what's gonna happen.
From Shanghai Noon, when Jackie Chan and Owen Wilson introduce themselves:
More like I'm Mr Literary Reference: In Simon Hawke's The Timekeeper Conspiracy Lucas gives his names to The Three Musketeers as "Alexander Dumas". After the action is over, the Temporal Corps erase him from their memories, as it wouldn't be good to have the author of their story finding his name in their accounts of it.
Merges with Beethoven Was an Alien Spy in the Star Trek: Deep Space Nine novel trilogy "Millennium". Near the end of the second book, Vash and alternate future versions of Nog and Picard go back in time to ancient Bajor to become the seers "Eilin", "Shabren", and "Naradim", who are famous in the future because of their predictions of recent events.
Granted, Nintendo existed at the time... As a card company. And Standard Games (later SErvice GAmes) existed too. Not that they were well known, or did business in the UK at all.
In the Tim Powers novel The Anubis Gates, a character adopts the alias "Humphrey Bogart" in early-19th-century London. A variation in that the character is not himself a time traveller, but picked up the name from another character who is.
The time travelers also whistle Beatles tunes as a way to recognize each other.
Backfires horribly in the 1632 series, when a character tries using this kind of name for misinformation. Turns out the person he was trying to fool had managed to buy a copy of the Encyclopedia Britannica from the late 20th century, as opposed to the 1911 edition that most others have acquired, and did not appreciate being told that his prisoner, Mr. James Bond, was engaged to a Marilyn Monroe, and they could expect a visit from a great gunmaker from the future, Elvis Presley. Whoops.
Played straight in 1635: The Papal Stakes, where Harry Lefferts uses the code name "Vulcan" when meeting with "Romulus". This actually makes sense, as any down-timer would assume that Romulus would be meeting with Remus.
Played with in Dream Park, when Griffin is pressed to explain in-character why his LARP character is "the greatest thief in the world". His first two examples are generic fantasy-thief tropes, but his third boast is that he owns the only surviving black-market copy of Star Wars. Other Gamers point out the in-story anachronism, as the South Seas Treasure game they're playing is set in the 1950s.
In a variation that involves an alien world instead of time travel, the Superman novel "Last Son of Krypton" by Elliott S! Maggin, has Lex Luthor use the alias "Abraham Lincoln", which Superman recognizes but which the aliens don't.
Diana Gabaldon's Outlander series has a scene where one time traveler uses the line "Does the name 'Ringo Starr?' mean anything to you?" as a shibboleth to recognize another.
In the Stargate SG-1 episode "1969", Jack O'Neill said he was "Captain James T. Kirk" before switching to "Luke Skywalker". This ignores the fact that Star Trek actually did exist in 1969, although the interrogating officer in the scene had apparently never heard of it. Which isn't too far-fetched.
This may have been intended as a subversion, with Jack switching once he realized it would exist back now.
The latter probably didn't work because a "Great and Powerful" Goa'uld wouldn't be flying around in a transport ship.
SG-1 and Stargate Atlantis would often do a variation where a reference or Phlebotinum Analogy would hinge on some pop culture reference to inevitably be lost on alien team members Teal'c, Vala, Ronon and Teyla. But subverted toward the end of the series as the aliens — especially Teal'c — became more familiar with Earth culture.
Teal'c in particular became an avid Star Wars fan. Which later created a joke when Vala (another alien) mentions how she got pregnant without having sex, asking if anyone had ever heard of such a thing. Mitchell and Carter both seem to think of Jesus (actually Mitchell's thinking of King Arthur) but instead it's Teal'c who exclaims "Darth Vader", prompting a worried Vala to ask "How did that turn out?"
None of those examples turned out well, of course, including Vala's own child.
In the Doctor Who episode "The Empty Child", set in 1941, Rose jokingly refers to the Doctor as "Mr. Spock" when pressed for his name. Captain Jack, who's from the distant future, doesn't recognize the reference, and Hilarity Ensues.
In "The Fires of Pompeii," when pressed for a name, the Doctor sputters "I am... Spartacus." His companion Donna leaps in with "And so am I." In the same episode, he also explains her bizarre personality with "She's from Barcelona".
In another Doctor Who episode, the Doctor shows up in the 19th century claiming to be Sherlock Holmes (and dressed the part). Unfortunately, the Big Bad's human assistant has heard of Arthur Conan Doyle and his novel. In fact, he has mentioned it in all but name earlier in the episode to a female Silurian detective, upon whom the character is actually based.
In Life On Mars, Sam goes undercover as "Tony Blair" at one point, with Annie as "Cherie." Later, when Gene tags along, he's given "Gordon Brown" as his alias.
Sam: This is my friend Gordon Brown, and his wife... uh... Gene: Suki.
Also, when his mother asks his name he says "Detective Inspector Bolan", a mere few scenes after he's met T-Rex's lead singer.
And in the US remake, Sam introduces himself to his mother in 1973 as Luke Skywalker. He says it's Navajo when she comments on how odd it sounds. In this case, he panicked when she asked for his name and that was the only thing he could think of.
Gene: What do you think I'm doing here, Tyler? Sam: Building a Death Star? Gene: What? Sam: Nothing.
Also in the US version, Sam uses the alias "Tom Cruise" while impersonating a pilot, and "Bono" while undercover in the Irish mob. On an undercover mission where Gene and a woman have to accompany him, he assigns them the aliases George and Laura Bush.
This Critical Miss article describes a group of role-players playing as time travelers gone to WWI Germany using the aliases Heinrich Himmler, Joseph Goebbels and Rudolf Hess. Which would have worked if not for other time travelers.
Those people did exist during that time, but they weren't famous yet. Which made them both sufficiently German-sounding (duh) and easy to remember.
Static Shockinverts this: Virgil travels to the future (specifically the Batman Beyond future) and sarcastically tells Terry he's Beyoncé when he doesn't believe him to be Batman. Terry apparently doesn't get it, since he introduces him to Bruce that way seconds later.
Subverted in the South Park episode "Trapper Keeper", which features a cyborgExpy of Star Trek and Terminator characters from a post-apocalyptic Bad Future who refers to himself as "Bill Cosby", a reference that was already over a decade old. Not only does he get away with it (especially given how white he is), but he also manages to pass for a 4th grade student despite looking and sounding like a homeless (the only aspect anyone does pick up on) and mentally imbalanced middle-aged man.
Stan: Ms. Crabtree, there's another creepy homeless person on the bus!
Ms. Crabtree: Sit down and shut up!
Stan: But they smell like pee!
VSM471/"Bill Cosby": I am not a homeless person! I am a fourth grade student! My name is... Bill Cosby.