The question is self-explanatory, but I want to make some remarks in order to prevent the responses from going off into undesirable directions.
It seems that every few years I hear someone ask this question; it seems to hold a perennial fascination for research mathematicians, just as quests for short proofs do.  The trouble is that it has strong urban-legend tendencies: someone will say, "So-and-so's thesis was only $\epsilon$ pages long!" where $\epsilon \ll 1$.  It will often be very difficult to confirm or disconfirm such claims, since Ph.D. theses are often not even published, let alone readily available online.  If you Google around for a while, as I did, you will find many dubious leads and can easily waste a lot of time on wild goose chases.  Frankly, I'm a bit fed up with this state of affairs.  I am therefore asking this question on MO in the hope that doing so will put this old question to rest, or at least establish provable upper bounds.
I would therefore request that you set yourself a high standard before replying.  Don't post a candidate unless you're sure your facts are correct, and please give some indication why you're so sure.  Read the meta discussion before posting.  (Note that the meta discussion illustrates that even a MathSciNet citation isn't always totally definitive.)  Include information about the content and circumstances of the thesis if you know it, but resist the temptation to gossip or speculate.
I'm not making this question community wiki or big-list because it should ideally have a definite answer, though I grant that it's possible that there are some borderline cases out there (perhaps there are theses that were not written in scholarly good faith, or documents that some people would regard as equivalent to a Ph.D. thesis but that others would not, or theses in subjects that are strictly speaking distinct from mathematics but that are arguably indistinguishable from mathematics dissertations).
Finally, to anticipate a possible follow-up question, there is a list of short published papers here (search for "Nelson").  Note that the question of the shortest published paper is not as urban-legendy because the facts are easier to verify.  I looked up the short papers listed there myself and found them to be quite interesting.  So in addition to trying to settle an urban legend, I am hoping that this question will bring to light some interesting and lesser known mathematics.

David Rector's thesis ("An Unstable Adams Spectral Sequence", MIT 1966) is 9 pages, according to the record at the MIT library.  I haven't seen the actual thesis for many years, but I'm pretty the actual mathematical content takes about 3 pages total, and is largely identical to the published version in Topology (1966, same title, doi link: https://doi.org/10.1016/0040-9383(66)90025-5), which is 3 pages plus bibliography.  (Dan Kan, his advisor, likes short papers.)