What is the real title of Shakespeare's play, A Midsummer's Night Dream or Midsummer Night's Dream ?

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 href="http://www.shakespeare-online.com/plays/midsummer/mdshistory.html">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.