The title is indeed A Midsummer Night's Dream. The clue is in the meaning of the possessive word. A dream is something that typically belongs to the night rather than to the time of year, midsummer. If the title of the play was A Midsummer's Night Dream, grammatically it would mean that the night belonged to, or was a part of, midsummer, rather than the dream being a part of the night that happens to fall on midsummer.
While it would make sense to say "a midsummer's day," meaning that the day is one that belongs to the midsummer season, it only makes sense to say "a midsummer night's dream," which shows us that the dream belongs to the night, which happens to fall on midsummer.
To confirm this, the title refers to the midsummer festival. Hence, the grammar shows us that the dream belongs to the night of the festival. We could also say "the festival night's dream," or "May Day eve's dream."
In addition, some editors, such as Katharine Lee Bates, have also hyphenated the title, A Midsummer-Night's Dream. This makes sense because midsummer night is really a compound noun preceding the noun dream, and therefore, may be hyphenated. If we hyphenate the title, we see that the dream belongs to, or is a part of, the midsummer night, rather than just night.
The correct title of the play is A Midsummer Night's Dream. (Note that, according to the Chicago Manual of Style, "Titles of plays, regardless of the length of the play, are italicized.")
One way to remember this title is to compare it to the title of another famous play, this one by Eugene O'Neil.
Long Day's Journey Into Night
A Midsummer Night's Dream
Like the word "long" in the first title, the word "midsummer" is simply a modifier of the word that follows and does not take the possessive form.