Enotes has a very nice succinct explanation of this title.
In order for the title to have any meaning for the contemporary student of Shakespeare’s play, its origin must be explained. At the time the play was written, only three seasons were observed: autumn, winter, and summer—which included what we now consider spring and began in March. Therefore, the play, whose action takes place on the eve of May Day, actually is in midsummer as Shakespeare knew it. This was the time of year when animals were traditionally let out to pasture and the spirits of nature were thought to be abroad.
The action takes place in the fairy wood, which may be what the “dream” part of the title refers, although it may refer to another common custom, the divining by midsummer dreams and flowers who one’s lover is or whether one’s lover is faithful, just as the characters in the play do.
I would add to this that it is suggested by some scholars that Shakespeare wrote the play for the wedding celebration of a member of the nobility, which gives a reference for the character Theseus, for example, being based upon someone Shakespeare actually knew in real life. It would not have been unusual for Shakespeare to have been commissioned to create a play, as this paid him money for his work, just as ticket sales at public performances did.
However interesting any reasoning as to why one of Shakespeare's plays is named what it is can be to consider, we cannot be completely sure why it bears the name it does. So, you should be careful to consdier possibilities as to why it is named A Midsummer Night's Dream with this caveat in mind.
Please also see the essay by Wolfgang Clemen linked below as an "Overview" of the play. He also offers a perspective on the meaning of the title.