What is the significance of Wemmick's museum in Great Expectations?
In Great Expectations, as in many of his other novels, Charles Dickens delights in creating humorous and whimsical characters with odd names. Wemmick is such a whimsical character, although at first he does not seem so as he is cryptic in the office of Mr. Jaggers as he speaks with his "post office" mouth that merely takes in information and sends it out without any of his own personality being revealed.
However, when Pip is invited by Wemmick to have supper at his home, this odd character created by Dickens serves as an example of the theme of Appearances vs. Reality. For, unexpectedly, indeed, Pip is brought to a world that seems completely out of character with the nondescript, coldly-business-like clerk of Mr. Jaggers. There, at his home, Wemmick exhibits warmth and love toward his "Aged Parent," as well as a creative spirit which entertains his old father. The ritual of firing the canon, Wemmick tells Pip, helps to "sweep away the Newgate cobwebs." So, after Pip visits the home of Wemmick, he gains a new perception of the man who becomes a valued friend.
Wemmick's little museum of mementos taken from criminals also points to the humanness of this dual character, and the differences between appearance and reality. When Wemmick conducts Pip through the prison where the criminals greet the clerk as a friend, Pip realizes that, although Wemmick seems wooden in the conduct of business in the office of Mr. Jaggers, he--unlike Jaggers--does not dismiss the criminals as mere matters of business, after all. That he keeps things of these criminals points to Wemmick's having perceived them as individuals.
When you mention his "museum," I assume that you are talking about his collection of mementos from various criminals. Things like the locks of hair, razors, etc.
To me, you could argue that these are significant in two ways.
First, they seem to tell us something about Wemmick's character. He is something of an eccentric when he is away from the office. The museum and his "castle" show us this.
Second, I think it sort of hints at the source of Pip's new wealth. It hints at the importance of crime to Pip -- the fact that all of his "great expectations" are coming from a criminal.