Some critics argue, although it is hard to confirm, that we can understand the title of A Midsummer Night's Dream in context of the play's events if we know that the Elizabethans still only recognized three seasons of the year ("Historical Background," eNotes). The three seasons were autumn, winter, and summer. The Elizabethans also still used the Julian Calendar, which placed the new year as starting in March, with New Year's Day on March 25th. Hence, according to some critics, the new year began in March, which was summer, although that is hard to confirm. If summer begins in March then May Day, or May 1st is certainly in midsummer as Shakespeare knew it.
We know that the play centers around May Day because when the lovers awake in the forest and are found by Theseus and Egeus, Theseus observes that the couples must be in the forest at dawn "to observe / The rite of May" (IV.i.132-133). Critics also observe that May Day was often celebrated in the Elizabethan era at sunrise with an opening ceremony of singing ("Historical Background," eNotes). Hence, we see, that according to some critics, the play is set beginning on the eve of May 1st and continues through to May Day, which was indeed considered to be midsummer in Shakespeare's time period. Therefore, the title A Midsummer Night's Dream is completely fitting.
The title of one of Shakespeare's greatest comedies "A Midsummer Night's Dream" is an allusion (a reference) to the summer solstice (the longest day of the year), around June 23, but in fact the play is set in May. Still, the holiday of "A Midsummer Night's Dream" is appropriate since this day was associated with magic (herbs gathered on this night could charm), with lover’s dreams, and with madness. The very essence of the word "dream" in the tile also alludes to the subconscious, the very part of us in which we dream, when we impart our desires for better or worse upon our subconscious to take form. That is what happens to some of the characters in the play in which spells are cast and their subconscious is brought forth bringing to the surface one's deepest desires, thus adding to the comical elements of the play.