Mercutio is using a pun here, which is one of Shakespeare’s favorite literary devices. A pun is the use of a word that can mean two different things in the context in which it is used.
To understand the pun, you need to be looking at the quotation in context. Mercutio, of Romeo’s house of Montague, has been sword fighting with Tybalt, of Juliet’s house of Capulet. Romeo wishes to avoid physical violence between the two families, since he is in love with Juliet. When he moves to separate Mercutio from the fray, Tybalt stabs Mercutio fatally under Romeo’s constraining arm.
For a few moments, there is what sounds like lighthearted banter between the Montague’s people—Romeo, Mercutio, and Benvolio. When Mercutio realizes he’s dying, he says:
Ask for me tomorrow and you shall find me a grave man.
The word “grave” creates the pun. It could mean that Romeo will find him to be a “serious” man tomorrow, or that he will find that he is dead and in the grave tomorrow. Mercutio means the latter, since he knows that he is dying.
It fits with Mercutio’s character in that he tends to be very excitable and demonstrative in his speech, and he has punned before in the play.
Mercutio is a funny, witty individual who continually makes jokes and puns throughout the play. Mercutio is also considered a rather clever skeptic. In Act 3, Mercutio duels with Tybalt and Romeo intervenes in an attempt to stop the fighting. Unfortunately, Tybalt mortally wounds Mercutio before fleeing the scene. When Benvolio asks Mercutio if he is okay, Mercutio insists that the wound is merely a "scratch." Mercutio curses both of the families and sends his page to grab a doctor. Mercutio, with his usual dry wit, comments,
"Ask for me tomorrow and you will find me a grave man" (3.1.94-95).
As was mentioned in the previous post, Mercutio's statement includes the literary device known as a pun. A pun is a humorous play on words which employs a double-meaning. The word "grave" could refer to Mercutio's solemn attitude or suggest his final resting place. Despite Mercutio's impending death, he remains witty and humorous which is typical of his character.