Besides proving new theorems, how can a person contribute to mathematics? There are at least a few things a person can do to contribute to the mathematics community without necessarily obtaining novel results, for example:


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*Organizing known results into a coherent narrative in the form of lecture notes or a textbook

*Contributing code to open-source mathematical software


What are some other ways to make auxiliary contributions to mathematics?
 A: Find problems to solve. These can be open mathematical questions but (more important) new fields of application that may also need new mathematical concepts. Calculus e.g. was founded because it was needed for physics. So any scientist who dares to show his problems to mathematicians helps mathematics as well. 
A: You can create new jobs for mathematicians, e.g. by funding institutes like Jim Simons. Arguably this does much more for mathematics than actually doing mathematics due to replaceability: the marginal effect of becoming a mathematician is that you do marginally better mathematics than the next best candidate for your job, which is a much smaller effect than creating a new mathematician job where there wasn't one before.
You can also work on tools for mathematicians to use like arXiv (or MathOverflow!). Arguably this also does much more for mathematics than actually doing mathematics. Incidentally, arXiv was developed by a physicist, Paul Ginsparg, and almost none of the mathematics graduate students I've talked to about this know his name. 
A: Even people with no mathematical background at all can contribute to mathematics.
One obvious way is by running software such as the GIMPS client for finding Mersenne primes, though the value of such primes to theoretical mathematics is debatable.
Another, vastly more important one is to contribute anything, literally anything at all to human civilization. In reality, it is that complex civilization which makes mathematics possible in the first place. Because they do not have to scramble for food in the dirt at the risk of their lives, people have the time to explore "non-productive" subjects such as mathematics. The number of great minds who could have become Eulers or Euclids but did not because they had to work as swineherds or died of a flu at age nine surely outnumbers those who actually did ever do any mathematical work. By keeping civil order, hospitals and agriculture going, you keep mathematics going as well.
A: There is a lot of work done towards formalizing existing proofs into logic software and proof wikis.  The formalization of a proof can be very helpful economically.  It can lead to more confidence in proofs and making searching for results more feasible.
(Mario here:) I am an undergraduate who works with the program and library Metamath for doing formal proofs. There is a TON of work that still needs to be done in formalizing even just the standard undergraduate curriculum, and it's all easily accessible to any reasonably bright undergraduate. It helps to be good at programming or at least thinking like a programmer, since the work you do looks a lot like programming to the uninitiated and formal math is just as unforgiving to typos as any compiler. But even though there is a lot of general problem solving involved, the path is all more or less written out by other mathematicians (as "hand proofs"), so the way forward is always clearly delineated and it might even be ungraciously called "transcription".
I'm going to try to keep it up in graduate school, but even just as a pastime it's good for the brain and gives you a great sense of accomplishment, in addition to making you understand the corresponding hand proof on a deeper level than pretty much any kind of study out there. It's not for the faint of heart, but for precision-minded amateur or professional mathematicians with a programming bent I could not recommend it enough.
A: Many problems of theoretical physics (e.g., string theory) are related to the development of some new math. This severely requires the services of able mathematicians. In the past, Riemann, Grassman, Hilbert, Poincare, Elie Cartan, deed so for the discovery of Einstein's equation in GR. Physicist Witten was awarded Field Award for his services in Math research.  
A: As @Hennobrandsma said, you can teach mathematics well.
In addition to developing tools which aid mathematics research (as you mention), you can also work to develop tools which aid teaching mathematics.  
Without coming up with "new mathematics", you can apply mathematics in novel ways to other fields.  I think this is especially true if you can find natural and really useful applications for some piece of mathematics which has not been applied yet.  This is the best kind of "advertisement" mathematics can get.
A: I would say that though obtaining results is crucial, introducing new concepts, new connections, or even new perspectives looking at classical mathematics sometimes are more important.
Gauss, for example, introduced the concept of congruence in number theory and, arguably, number theory has then been developed systematically. The introduction of irrational numbers also solves old problems such as squaring a circle.
Einstein, for instance, found the connection between gravitation and curvature, so that differential geometry has been involved with physics. Kolmogorov, for another example, noted the connection between probability and measure theory, so that we have the modern theory of probability.
Klein, for example, proposed to view the geometries from the group-theoretic point of view and contributed a lot to modern geometries. Hilbert, for another instance, looked through the nature of mathematics and rigorously treated mathematics axiomatically.   
Personally, I believe that these contributions to mathematics are of higher form of contributions.
A: Another way to contribute that hasn't been mentioned yet is advocacy. It is important to raise awareness on the importance of mathematics and scientific literacy in general. For instance, here are a few concepts that it would be useful to disseminate to the general public:


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*that some knowledge of mathematics, statistics and science is important for the average person, too.

*that there is still active research in mathematics; theorems weren't all discovered 300 years ago.

*that mathematics (and STEM in general) is a viable career path, and people interested in it shouldn't be laughed at.

*that research, even basic research, is an important endeavour and needs funding.

A: It is useful to contribute to databases of mathematical objects such as Sloane's Encyclopedia of Integer Sequences, FindStat and others.  These resources have become invaluable for mathematicians searching for references to objects that they don't know the name of.  Using these databases, mathematicians are able to find connections between seemingly distant fields if both make use of a certain integer sequence or other object in a database.  For more information, see Billey and Tenner's manifesto here.
