We cite tweets a lot here at WebProNews because, one, we cover social media-related news, and two, it's how a lot of information breaks sometimes. Sometimes they're funny, sometimes they're means to convey information, sometimes we're just checking the pulse of the laity when something is announced. Beyond that, Twitter has also become one of the primary outlets through which famous and notable people dole out exclusive information or make announcements.
The Modern Language Association seems to have recognized that writers' best original source for information sometimes may come from Twitter updates and have therefore devised a standard rule for properly citing a tweet in an academic paper.
And yes, if you read between the lines there, you're moving right along with me: tweets are officially an acceptable source in academic and peer-reviewed papers, at least as far as the MLA is concerned.
The MLA's website explains how to correctly cite a tweet in a paper:
Begin the entry in the works-cited list with the author’s real name and, in parentheses, user name, if both are known and they differ. If only the user name is known, give it alone.
Next provide the entire text of the tweet in quotation marks, without changing the capitalization. Conclude the entry with the date and time of the message and the medium of publication (Tweet). For example:
Athar, Sohaib (ReallyVirtual). “Helicopter hovering above Abbottabad at 1AM (is a rare event).” 1 May 2011, 3:58 p.m. Tweet.
The date and time of a message on Twitter reflect the reader’s time zone. Readers in different time zones see different times and, possibly, dates on the same tweet. The date and time that were in effect for the writer of the tweet when it was transmitted are normally not known. Thus, the date and time displayed on Twitter are only approximate guides to the timing of a tweet. However, they allow a researcher to precisely compare the timing of tweets as long as the tweets are all read in a single time zone.
In the main text of the paper, a tweet is cited in its entirety (6.4.1):
Sohaib Athar noted that the presence of a helicopter at that hour was “a rare event.”
The presence of a helicopter at that hour was “a rare event” (Athar).
College students around the world just collectively sighed.