Quite a bit has been written about this title, and I suppose there are several subtle and surprising things that can be gleaned from it. There are a few clues, linguistically, that the title gives us, and certain pointers from history that can help.
Midsummer in England during Shakespeare's time (right now, in fact, around the summer soltice, is what they called "midsummer", and midsummer eve was a time of magic and enchantment in the pagan calendar and still believed in long into Christian times) was a time for feasting and enjoyment -- and for getting married! Then, as now, June was favored for weddings (partially because it was after both the Lent and Easter seasons of the church, when weddings were not performed as often, and also after the month of May, the Virgin Mary's month, when not as many weddings were perfomed because the month was supposed to be dedicated to the honoring of Mary), and the festival atmosphere of midsummer in the late-medieval and Renaissance times was conducive to romance. These connotations could lend themselves, quite easily, to comedy -- and especially a comedy surrounding romantic love and marriage. Midsummer implies that this is a time --and a play-- of certain frivolity and gaiety, which this play certainly is.
Also, it was believed that certain plants were magically healing, especially when picked on this night. This ties in with the whole drug/love potion theme of the play. There are also the enduring fertility myths surrounding this time, which of course relate to the subject matter of this play.
The "Dream" part of the title has certain easily discerned meanings. The whole story of the fairies is the supernatural, and therefore related to dreams. It could be argued that Bottom really only dreamed his whole night with Titania, the queen of the fairies, and that it could not possibly be real. Also, the action of the drug on the people who it was given to made them see and feel things that were dream-like -- in only certain cases they were lasting (like Demetrius). Also, the play the rude mechanicals perform can be likened to a dream (in that it is not real, but only a visual show, such as a dream is).
The fact that Shakespeare titles it "A" Midsummer Night's Dream implies, perhaps, the trifling nature of this comedy. Compare "The Tragedy of Hamlet, Prince of Denmark", with its definite tone, to the relatively innocuous title of this play. It is only "a" dream -- not one dream, an important dream, or even "the" dream. It is a play of lightness, humor, and imagination, and the whole play can be, even with the Hippolyta and Theseus and Rude Mechanicals stories, thought of as one person's dream after a night of merry-making.