When Mercutio says, "Ask for me tomorrow, and you shall find me a grave man," he is indeed making a pun on the word grave, which means both serious and a hole in which a dead body is buried. I think such a pun, usually a comedic device, illuminates his character even further because it comes at a markedly not comedic moment in the play. Mercutio has just been stabbed and knows that he is about to die, and yet, for some reason, he is still making jokes.
All along, it seems that Mercutio has been unable to deal with strong feelings, either in others or in himself. When Romeo is depressed because Rosaline does not return his love, he doesn't want to go to the Capulets' party and have fun because feels as though he has a "soul of lead." Mercutio lectures Romeo on the nature of dreams, saying that they are "the children of an idle brain, / Begot of nothing but vain fantasy." He cannot recognize the strength of Romeo's feelings.
It happens again when Romeo runs off into the night to look for Juliet after the party. Mercutio calls for him, mocking him the entire time: making light of love in general and Romeo's love (of Rosaline) in particular, even using very sexual and bawdy talk. Mercutio makes love all about sex, failing to recognize the strength of his friend's feelings for Rosaline (or Juliet). In fact, he makes so many sexual jokes during the play that he begins to seem incredibly immature, emotionally speaking. I think we can look to his "grave" pun in act 3 as further confirmation of this: he has failed to take responsibility for his own part in the violence with Tybalt, he makes jokes, and then he blames the two families for his death, which was, in many ways, his own fault.