How many papers should a PhD student in math try to publish before graduating? I am wondering how many papers does an average academia-bound PhD student in math try to get published? I want to eventually work in a research-oriented university rather than a teaching-oriented university. Obviously the more good mathematics one gets published the better, but what would be average for someone who goes on to work as a professor at a research university?

I am particularly interested in the field of combinatorics.
I agree entirely with the answers of Mark and Brian.  Since my name has been uttered three times and I've been summoned here already, let me add a few nuances.

1) Asking for a number is certainly the wrong question.  What you want to show as a graduating PhD student seeking an academic career is substantial results and the promise of more (and better) such results to come.  Multiplicity is not the point at this stage of the game.  Better one very good result than five pretty good ones.

2) Zero is a very interesting number nowadays.  It corresponds both to people who have not done any publishable work (note to academics in STEM fields other than mathematics: the standards for publication in a non-predatory math journal are quite high; the most common number of publications for a PhD in mathematics is certainly zero, and many PhDs are awarded for theses which could only be publishable with a significant amount of additional work which the PhD candidate is almost certainly not going to do) and future stars at elite places who do not need to show their partial work because their superstar advisors will tell the story for them.  So zero publications is very common...but it's most commonly bad.  

3) Publication quotas have been rising dramatically in mathematics, even in the last dozen years.  I wrote elsewhere that I got my PhD in 2003, my first publication was in 2005, and I did well on the tenure-track job market in 2006 with five accepted papers.  Well, it's true....but that story is so 12 years ago.  Most postdocs we have hired at UGA in recent years have had some accepted papers or at least an arxiv preprint that the interested parties can look at and convince themselves is going to be accepted.  Also at UGA we have numerous, highly systematized opportunities for graduate research, which results in a substantial number of publications co/authored by graduate students.  We also give 2-3 graduate research awards per year, and these awards are usually given for preprints or publications, not just "promising thesis work".  All three of my graduated / soon to graduate students have had arxiv preprints or publications by the time they went on the job market; two out of the three had more than one.  In the limit, Princeton students will get five years of postdocs and be able to release their Annals paper while they're on the tenure track, and everyone else who's serious will have some tangible product upon graduation.  Well, obviously that's an exaggeration, but there's some truth to it.

4) It is true that numbers of publications are sensitive to the subfield.  They're sensitive to a lot of other things as well -- e.g. the length and quality of the publications -- but if we want to sling generalities: applied math people publish way more than pure math people (and they cite each other with enough frequency to make the stereotypical pure mathematician want to roll her eyes every time she hears "impact factor"); they may even publish some of their good papers in conferences, which is almost unheard of in pure math.  In pure math, combinatorics is one of the highest publishing fields, and the papers tend to be shorter (and often lighter, but even a weighty breakthrough paper is probably going to be short compared to other fields).  Algebra and number theory publish a bit more than average, generally.  Analysis is a broad field and seems roughly in the middle.  Geometry depends sensitively on the subfield: discrete geometers publish a lot; differential geometers less; algebraic geometers really less.  Topologists seem to publish very few papers.  Well, it was fun to play but I wouldn't take any of that very seriously (though argue if you want, that's part of the fun).