Literary critic Northrop Frye points out that the Elizabethan calendar still only recognized three seasons in the year, "summer, autumn, and winter" ("Mythological Background"). He further states that summer included spring and began in March thereby placing May in the middle of summer and May Day as the mark of midsummer.
We know that most of the action of the play takes place on May Day eve because, the morning after the night in the woods, Theseus finds the four lovers asleep in the forest and remarks:
No doubt they rose up early to observe The rite of May; and, hearing our intent,
Came here in grace of our solemnity(IV.i.132-133).
Hence, if the play is set between May Day eve and May Day, then we know that the action of the play indeed takes place in midsummer.
Frye also points out that May Day eve was traditionally recognized as one of the "haunted 'eves'" of the year, the first being All Hallows' Eve. May Day eve was a night on which Elizabethans believed that spirits were around, and those spirits could be either "benevolent or malignant" ("Mythological Background"). Hence, it makes perfect sense that Shakespeare made use of both the mischievous actions of fairies and the beneficial actions, as we see through both Puck and Oberon.
Thus, the title A Midsummer Night's Dream is fitting because the action of the play takes place in midsummer, and May Day eve was also believed to be a haunted night.