Download the latest FontAwesome.otf from the repo and put it
in ~/texmf/tex/latex/fontawesome-custom.
Create ~/texmf/tex/latex/fontawesome-custom/fontawesome.sty with this content:
\ProvidesPackage{fontawesome-custom}[2015/01/12 My local customizations to the fontawesome package]
\RequirePackage{fontawesome} % load the package we're modifying
\expandafter\def\csname faicon@stackexchange\endcsname {\symbol{"F16C}} \def\faStackOverflow {{\FA\csname faicon@stackexchange\endcsname}}
If you want to add a different new icon, have a look here and here.
You can use the icon like this in your tex file:
\documentclass{moderncv}
\usepackage{fontawesome-custom}
\moderncvstyle{classic}
\moderncvcolor{green}
\title{CV}
% Contact data
\firstname{Matthias}\lastname{Braun}
\extrainfo{\httplink[\faStackOverflow~Stack Overflow: Matthias Braun]{stackoverflow.com/users/775954/matthias-braun}}
\begin{document}
\makecvtitle
\section{Job experience}
\cventry{March 2013}{Trip to the moon}{}{}{}{}{}
\end{document}
Compiling with lualatex or xelatex produces this:

:-)Modifying a file managed by a Package Manager is not the best idea. A better way would be to make your own local package containing the customizations. You can see an example of how to do that here. – Paul Gessler Jan 12 '15 at 16:03.styintoTEXMFHOME, and put the updated font either there or in~/.fonts. Note that the package is LPPL which means you definitely should rename it if modifying it, in any case. – cfr Jan 12 '15 at 16:51