The title suggests an atmosphere of fantasy, whimsy, and imagination, which is a pretty accurate description of the magical wood where characters experience events that seem more like a dream than reality. Poor Bottom can't even begin to describe what's happened to him in the wood: "I have had a most rare / vision. I have had a dream past the wit of man to say what / dream it was" (4.1.214-216).
Shakespeare also knows that, after watching the play, we, the audience, might also experience some uncertainty about the difference between reality and illusion. (This is why Puck invites us to think of the play as a nothing more than a "dream" during the Epilogue. Go to "What's Up With the Ending?" for more about this.)
The title is also a pretty obvious shout-out to Midsummer's Eve (June 23), or the summer solstice. Elizabethans would have heard this title and thought "party time!"—in Shakespeare's day, Midsummer's Eve was all about celebrating fertility (not just the successful planting and harvesting of crops, but also the kind of fertility associated with dating and marriage). It was an excuse to party outdoors and the holiday involved dancing, drinking, and collecting medicinal herbs. For a lot of partiers, Midsummer's Eve was also supposed to be a time of mystery and magic, when spirits ran around causing mischief and teenage girls had dreams about the guys they'd eventually fall in love with and marry.
Our point? Shakespeare's title captures the festive vibe of the play and even enacts some of its rituals.
While we're on the subject of festivities, we should point out that Shakespeare also works some May Day festivities into his play. Remember when Theseus stumbles upon the sleeping Athenian youths in Act 4, Scene 1? He thinks they're passed out on the ground because they got up early and went into the wood to "observe / The rite of May" (4.1.137-138). (Note: The rites of May—games, festivities, etc.—were performed throughout May and June, not just on May 1.) "Maying" involved going into the woods in the early morning to gather up blooming tree branches (for decoration) and putting up "Maypoles" to dance around. In the play, Lysander mentions that he once met Hermia and Helena in the wood to "do observance to a morn of May" (1.1.169). May Day revelers also celebrated with big feasts and even elected a "Lord of Misrule" to preside over the festivities. Check out Puck's "Character" page to learn about how he fits the role of a Lord of Misrule.