Not to mention Ken Bone! 
Elizabeth Smart.
I lived in Sioux City briefly, a few years after Al Haynes landed Flight 232.
Rodney King
Kato Kaelin
Sidetrack:
The TV movie based on the crash (“A Thousand Heroes,” also known as “Crash Landing: The Rescue of Flight 232”) is, hands down, one of the best TV movies ever made–outstanding cast, non-exploitative handling of the material, a total winner; a must see!
I think Anne Frank wins. High school students today don’t know Lewinsky. Everyone knows Anne Frank. There are schools named after Anne Frank.
Hell, even Justin Beiber himself even hoped that she would have been a ‘belieber’ if it weren’t for that whole Nazi unpleasantness. That is famous right there.
Dr. David Dao
William Hung (American Idol “She Bangs”)
I’d agree with that. I must have missed seeing her mentioned.
John Wayne Bobbitt
Ben Linder, though 23 people would recognize the name anymore.
Henrietta Lacks is becoming well known lately, although many scientific types already knew who she was.
Still, I’ll have to cast my vote for Ann Frank.
Lenny Skutnik
Plane crash, Potomac River, winter, 1982. He jumped into the river to rescue a woman who was unable to grab onto a life preserver. Somebody I’ve never forgotten.
Elian Gonzalez
Christa McAuliffe, the schoolteacher killed in the Challenger space shuttle disaster.
I’m not sure that Monica Lewinsky really counts. She’s famous for her interactions with a very famous person, and she knowingly and deliberately chose to engage in those interactions. Sure, she never intended for it to become known to the general public, but she also had to know that if they did become known, she’d be famous for it.
Anne Frank, however, was (so far as she knew) just another teenage girl with a diary, who through no fault of her own happened to be alive during a horrific episode of human history. She definitely counts, and is going to be hard to top.
Alan Kurdi. 
His death caused the fall of at least one government.
I think Anne Frank wins, cue to the combination of how extremely famous she is, and the utter mundanity of the action she took that led to her fame.
A fair number of the people mentioned might have been unknown to start with, but did make the conscious choice to take an action which had, as a reasonably possible outcome, a chance of leading them to prominence. Ken Bone, for instance, did ask a question at a nationally televised political debate. It was a surprise that he became as much of a meme as he did, but that’s different from someone just walking down the street and then BAM. Same for David Kaczynski, Joe the Plumber, Christa McCaulliffe and Monica Lewinksy.
There’s also an entire category of people who were involved in brief-media-craze stories, such as John Wayne Bobbitt and Elian Gonzalez, but for the most part, their fame has proven to be fleeting. (To that list we should certainly add the fantastically-named Joey Buttafuoco.)
People mentioned who I had not heard of at all before this thread (at least by name):
Erwin Kreuz (Tried to take a cab across the country, famous in the 70s for some reason)
Alexander Selkirk (Real-life Robinson Crusoe)
Lisa Gherardini (Mona Lisa named after her, Wikipedia claims it is now definitely known that she was the model for it, although that was controversial for years)
Jeremy Meeks (attractive criminal, became meme)
Ted Williams (sang on youtube?)
Baylee Almon (baby girl in iconic Oklahoma City bombing photo, died)
Greta Zimmer Friedman (Nurse in the V-J day times square kissing photo)
Harry Truman (Died in Mt St Helens)
Spencer Elden’s penis (he was the baby on the front cover of Nirvana’s Nevermind)
Dr. Kent Brantly (Survived Ebola)
Al Haynes (Landed an airliner in a crisis)
Dr. David Dao (The guy who was quite recently removed from a United plane, causing an incident)
Ben Linder (killed in Nicaragua by Contras, sparking international outrage)
Lenny Skutnik (He is notable for an act of heroism which led to him being mentioned by President Ronald Reagan in the State of the Union Address, which coined the term Lenny Skutnik referring to notable people who are invited to sit in the gallery at a State of the Union address or other joint meeting of Congress.)
Alan Kurdi (dead little boy in famous refugee photo)
Some of those I hadn’t heard of at all before this thread, regardless of whether I recognized their name:
Erwin Kreuz
Lenny Skutnik
Ben Linder
Al Haynes
Best known? In terms of average public recognition this guy will beat all your others by a mile even Anne Frank.