It would not be a good idea to modify the fonts in the MathJax distribution (unless you also rename them both in terms of the file name and the metadata within the file). The HTML-CSS output jax is set up to use a locally installed version of the fonts, if there is one, and so even if you modify your web-based version, you may not see the change if you have the original installed locally. Also, it is a very bad idea to have fonts with the same name but different characters.
In addition, you would need to provide not just the WOFF versions, but also the OTF and EOT versions of the fonts, as these are used by some browsers.
Finally, you would have to modify the data files used by MathJax to store the bounding box information for all the glyphs in the fonts.
So this is not a good approach.
A better one would be to create a new font with just the one new glyph that you need (or several fonts, if you need it in more than one style, as your message suggests you do), create the associated data for them, and insert that into the output jax's font data tables at the appropriate places. This is non-trivial, but could be done.
Davide