What kind of economic freeware exists for economics students? 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.