“What’s in there?”
As others have already pointed out, the actor who played Archie in TSOT is credited as “Billy” in TAB. As we know, all the Billy/Williams in the show = Sherlock (first name William), so we can assume that ‘Billy’ represents the child version of Sherlock in his own mind. Which seems like a fairly deliberate move, and made me think of the first scene we see ‘Billy’ in. John is carrying a hat box, which from the case details we can infer probably has a body part of the ‘dismembered country squire’ inside of it, likely his head. ‘Billy’ ( Sherlock ) asks what’s inside the hat box, and John tells him “Never mind” in an Eyes Wide sort of Alarmed way that suggests that whatever is inside is not for kids.
We’re then later shown John putting the hat box down on what is normally the kitchen table inside of 221B.
Not to now tell you that it’s All About Sex, but that’s where I’m going with this. Allow me to explain: Sherlock sees himself as a child trying to get a peak at John’s package. Alternative phrasing: Sherlock tries to get a head from John. Ima try again: Sherlock is interested in the head John is carrying at waist level. No matter how I slice it, the subtext seems to read at the very least a bit sexual to me, even just from John’s reaction alone.
However! John stops Child Sherlock from looking, and acts like he shouldn’t look. If whatever is inside the hat box was gory, it seems strange that Sherlock’s mind would stop his Archie Child Self from looking, as we know Sherlock bonded with Archie specifically over their mutual love of gore. So, my interpretation: Sherlock can’t have a relationship with John unless he does some growing up, likely in the form of facing some things about himself. He is barring this from himself (as MP John is really him at the end of the day). This fits, as a lot of the rest of the episode is spent exploring Sherlock’s complicated relationship with the side of himself that Moriarty represents, which includes his (homo)sexuality.
This is simplistic and vague because anything else is too speculative, especially as I don’t know for sure what is in the hat box. For example, maybe there is a head in the hat box and it represents Sherlock’s head, and this is meant to say that John (his heart) is ruling his head, but he can’t know that just yet. Or, the actual contents of the hat box are the ‘Ghosts of His Past’ the shadows that define etc., such as a repressed memory that Sherlock has had since childhood, which he has since given to MP John for safe-keeping. It could also just be a Blow Job Joke as I said above, and Sherlock may still think of himself as a child (he’s called himself a baby as recently as TSOT), and in his mind, John won’t be with him sexually until John stops seeing him as a kid. It’s pretty sad if Sherlock actually thinks this, seeing as John is ready to go at any time lol However, either way, I very strongly suspect that whatever is inside the hat box, it is linked to Sherlock’s ability to enter a sexual relationship, as John places it safely where they eat (food = sex metaphor) in 221b (the physical representation of Sherlock’s heart and love for John).