Filter design / Analysis apps
I believe that Octave and Python/Scipy are the most commonly used free software packages for digital signal processing (including filter design). They offer of course much more than you might be looking for. On the other hand, if you want to play around with the coefficients and get some quick feedback, it might be very helpful to have one of these (or similar) packages because then you can easily modify the filter, plot frequency responses, and even write your own little programs if necessary. I've used Octave for that purpose and for me it worked fine. If you use Octave, you'll need to install the signal processing toolbox.

A totally different approach would be to choose your favorite programming language and write some simple routines yourself. This is perfectly feasible for some common FIR design methods (such as windowing). In combination with some plotting software this may be all you need. In this case I would consult this document on digital filtering. This latter approach is of course not the most efficient one because you'll be re-inventing the wheel, but in this way you'll learn how it all works and you will not just be using a black box design method.