So somehow Sherlock managed to find out the missing word for somebody is actually anyone. How did he do that? What evidence was he able to obtain that enabled this discovery?
As far as I could tell, the only apparent "evidence" was Mary and Eurus (in disguise as Faith) saying the word anyone. But this isn't enough to lead Sherlock to the truth, is it?
There's a cliche in movies, TV, books, pretty much anything, where a detective-type character will experience a revelation from information they already know, but at a different perspective.
When Faith tells Sherlock that he's nicer than "anyone", Sherlock realizes that is the impossible one-word name of the person Culverton was going to kill. This is shown technically by Sherlock going through different times he heard the word "anyone" through an entirely different camera lens, or strange and unusual camera angles.
Well, we can't know how exactly Sherlock deduced that. He is very smart. Faith's memory was hazy, that's why she asked Sherlock to find out exactly who her father needs to kill.
After sometime she thinks that her father's confession was also followed by one word, that means the person he needs to kill is known on a first-name basis.
Moreover, Sherlock kept hearing the word “anyone” all around him and figures out that the name of the person Smith wanted to kill was anyone, leading him to believe that he was a serial killer.
Specifically, he hears this word from Faith...
FAITH: You’re not what I expected. You’re ...
SHERLOCK: What ... what am I?
FAITH: Nicer.
SHERLOCK: Than who?
FAITH: Anyone.
... Mary in her recording...
MARY’s VOICE: Don’t think anyone else is going to save him, because there isn’t anyone.
and Molly.
MOLLY’s VOICE: ... if you were to come round asking after him, that he’d rather have anyone but you.
MOLLY: Anyone.
Perhaps, this helped Sherlock to find out the missing word.
It's tempting to reason that Eurus manipulated Culverton into killing "anyone" and pointed him toward Sherlock as being a candidate.
Eurus also manipulated Sherlock into taking on her "case" which resulted in the confrontation she was after.
So, the notion of "Anyone" would be sourced by Eurus.
