You'll first need a webserver. Since you're running linux, the first and most obvious answer is Apache httpd. Another good choice here may be Lighttpd.
Now you need a way to serve dynamic content. My personal go-to here is PHP. Often times you can install Apache+PHP with just a couple packages. It looks like Lighttpd supports PHP, too.
This link will explain CGI, Perl, etc. What is CGI?
Finally, you need a way for the web application you develop to affect the rest of your black box. SQLite makes for a nice lightweight way to store data in an organized fashion. Your PHP web app could use the same database that the C programs read their parameters from! Another option would be just simply reading and writing to flat text files.