Can anyone recommend some free software for students studying economics for use in senior projects, theses, or dissertations?
I never understood why discussion about specialized software should be off-topic in the specialty's website. And of course I don't agree. So:
Before diving into R, which indeed appears to be the dominant (and rich) freeware for statistical computing, one can try Gretl. It is an Econometrics freeware, with a lot of functionality, a very good Random Number Generator, and has both menu-driven implementation but also code-writing by the user. It is easier for beginners - but it is serious stuff. The wikipedia page for Gretl lists some reviews about Gretl.
The other kind of "economics" (not mathematics) software apart from econometrics packages would be specialized simulation software, for say Dynamic Stochastic General Equilibrium Models, or micro-applications. For DSGE, one such is Dynare. A lot of DSGE-code for various other software can be found at International Network for DSGE Modeling, Monetary and Fiscal Policy.
Python is also something you could look at for econometric and/or data analysis in addition to R and Gretl mentioned above. Alternatively you could check out the Ox programming language. The Ox console is free to download for academic use. The Ox programming language is the basis for OxMetrics and it is relatively easy to use and extremely fast.
For optimization etc. you can get Octave which is a free alternative to MATLAB. I haven't tried Octave so I cannot comment on its qualities but its free so it might be worth checking out. You can also use Octave to simulate economic models in.
For simulation of economic models I would recommend you to check out Dynare which is a module for simulating and estimating DSGE and OLG models. Note that in order to use Dynare you'll need to have MATLAB or Octave installed on you computer (Octave is free!). I have used Dynare with MATLAB and it is really easy and intuitive to use.
For typesetting you should check out LaTex or the "easier-to-start-to-use" alternative Lyx. In case you want a "Word" type enviroment instead you should check out Open Office. Open Office also has a spreadsheet á la Excel.
There's a difference between "free software" and "free software for students studying economics". For example, when I was studying economics at University, the University provided (free) access to software such as Maple, MATLAB, Datastream, and Stata, to mention just a few.
These software packages are not free, but to economics students at many Universities, there is free access to all of this software making it free, effectively (to the economics student).
As a result, I'd recommend that economics students consult with their instructors to find out what software is being made available to them. The University doesn't pay the licence fees just for kicks.