This is addressed explicitly in the biography First Man: The Life of Neil A. Armstrong, by <PERSON>, which the film is based on, and whose author is presumably the <PERSON> mentioned in <PERSON>'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," <PERSON> would declare. "At least not that I can remember." As for <PERSON>, 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, <PERSON> took nothing else for family members--not even for his two boys, a fact that still distresses <PERSON>. "I assumed he had taken things to give to the boys later, but I don't believe he has ever given them anything. 573
<PERSON> can be thoughtful, but he does not give much time to being thoughtful, or at least to expressing it."
Another loved one that <PERSON> apparently did not remember by taking anything of hers to the Moon was his daughter, <PERSON>. 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 <PERSON>, just before he left the lunar surface on Apollo 17, had written the initials of his nine-year-old daughter, <PERSON>, in the dust. <PERSON> carried photos of his children to the Moon. <PERSON> left a picture of his family on the surface.
What if <PERSON> did something for <PERSON> but never told anyone about it, not even <PERSON>, 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 <PERSON>'s sister <PERSON>, who knew her brother as well as anyone.
"Did he take something of <PERSON> with him to the Moon?" was <PERSON>'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 <PERSON> living sons to be certainly relevant.
I'm bothered by the exact textual match between <PERSON> and <PERSON>'s report of June <PERSON>'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.) 715