Archie was a page boy at the wedding of John Watson and Mary Morstan.
Before the wedding ceremony, he spoke to Sherlock Holmes to be admitted to his role. However, Archie ends up questioning him about his work as a detective, and wants to see some photos of crime scenes, to which Sherlock acquiesces.
When Sherlock's best man speech is interrupted by a possible murder, Archie tells Sherlock that it was the 'Invisible Man' with the invisible knife, the one that killed the guard.
In The Abominable Bride, he is a minor character, who seems to assist Sherlock and John, as children would, and Sherlock's positive opinion does not appear to waver, despite the time difference.
Sherlock seems to like Archie and might be intrigued by the boy's cool demeanor. This is supported by how Sherlock is admittedly not good at remembering names, and seems to actively avoid interacting with children, yet makes an exception to both with Archie.