Did Neil Armstrong really do this on the Moon?

There is no evidence to show he did this, but it's not implausible.

I've read a number of historical accounts of NASA through the Mercury, Gemini and Apollo era, as well as biographies of many Astronauts (though not including Neil) and Engineers on the program and I've never heard of this before.

This is covered in an article in the Washington Post.

That said, it would almost certainly have been possible. Astronauts were allowed some personal items on the missions. Buzz Aldrin took items to take Communion on the Moon on the Apollo 11 mission. On Apollo 15 there was a scandal when the astronauts took a number of postal covers on the trip to sell for a profit later.

So, it's a fictional element of the story and there's no account written about it, but its not impossible or even improbable that he would do something like this.

This is addressed explicitly in the biography First Man: The Life of Neil A. Armstrong, by James R. Hansen, which the film is based on, and whose author is presumably the Jim mentioned in wcullen's answer.

Armstrong never released any information about the contents of his [Personal Preference Kit]. He agreed to do so for publication in this boo, but reported that he was unable to find the manifest among his many papers.

[...]

"I didn't bring anything else for myself," Neil would declare. "At least not that I can remember." As for Janet, the only thing taken to the Moon for her was the olive branch pin. "He didn't ask me if I wanted to send anything."

Perhaps surprisingly, Armstrong took nothing else for family members--not even for his two boys, a fact that still distresses Janet. "I assumed he had taken things to give to the boys later, but I don't believe he has ever given them anything. Neil can be thoughtful, but he does not give much time to being thoughtful, or at least to expressing it."

Another loved one that Neil apparently did not remember by taking anything of hers to the Moon was his daughter, Karen. What could have made the first Moon landing more meaningful "for all mankind" than a father honoring the cherished memory of his beloved little girl, by taking a picture of the child, dead now over seven years (she would have been a ten-year-old), one of her toys, an article of her clothing, a lock of hair, her baby bracelet? Astronaut Gene Cernan, just before he left the lunar surface on Apollo 17, had written the initials of his nine-year-old daughter, Tracy, in the dust. Buzz Aldrin carried photos of his children to the Moon. Charlie Duke left a picture of his family on the surface.

What if Neil did something for Muffie but never told anyone about it, not even Janet, because it was of such an intensely personal nature? How much more would posterity esteem the character of the First Man? It could have elevated the first Moon landing to an even higher level of significance. Among those who feel so are Neil's sister June, who knew her brother as well as anyone.

"Did he take something of Karen with him to the Moon?" was June's rhethorical question.

"Oh, I dearly hope so."

Perhaps the mystery will be solved when humankind returns, as it surely will, to Tranquility Base.

As mentioned in the existing answers, this makes the incident a plausible educated theory, but I do find the added context w.r.t. Neil's living sons to be certainly relevant.

I'm bothered by the exact textual match between Hansen and Singer's report of June Hoffman's response. Were they both present in the same interview? Did she give a word-for-word exactly identical response on two separate occasions? Is either of the two accounts a paraphrase of the other? (Just things to consider.)

Possibly

Apparently screenwriter Josh Singer

asked Armstrong's sister, June Hoffman, whether her brother had left something for Karen on the moon. June's response: "Oh, I dearly hope so."

"Nearing the bar we were setting for ourselves in terms of accuracy, I never would have felt comfortable doing that on my own," Singer said. "But hearing that it's good enough for Jim, and good enough for June, it was good enough for me."

So, it wasn't fully a Hollywood-ized version of the moon landing. It is more of an educated theory.

https://www.esquire.com/entertainment/a23744321/first-man-fact-fiction-moon-neil-armstrong-daughter-bracelet-real/

Armstrong never spoke about this publicly.

Jay Barbree's biography Neil Armstrong: A Life of Flight (Thomas Dunne Books, 2014) says, on p. 272 of the paperback edition,

In the lunar dust, Neil and Buzz placed mementos for the five deceased American and Russian space flyers, Gus Grissom, Ed White, Roger Chafee, Vladimir Komarov and Yuri Gagarin (the first in space had died in a plane crash the year before), and one small cargo—private and honorable—carried by Neil. It was not to be divulged. It was a diamond-studded astronaut pin made specially for Deke Slayton by the three Apollo 1 astronauts who planned to fly it on their mission before that dreadful fire. And there was one other remembrance. Very special and dear to Neil, a part of an unfinished life he so wanted to leave on the moon, and he did.

It's quite clear that the “unfinished life” is that of his daughter Karen Anne (“Muffie”). Barbree also writes (p. 44),

Neil seldom spoke of this overwhelming heartache in his life. But others were convinced Karen Anne's death was the single most important reason he would submit his name to become an astronaut. Her death gave him a new purpose. A few months before Neil's own passing, I asked him, “Is there something of Muffie's on the moon?”

I read his smile to mean yes.

Barbree was NBC's space correspondent basically forever and reported on every manned NASA launch. He describes himself as being a close friend of Neil Armstrong's for over 50 years, and the introduction to the book talks about how Armstrong was a stickler for accuracy. Barbree is particularly sensitive about Karen Anne's death, as one of his own sons died during childbirth.