What is the best Linux software to monitor web traffic on router?

 am assuming that you want to log and monitor the traffic which is passing through a hardware router.

Your question seems incomplete to me because at router level, it is difficult to decide if data which is being transferred is for web or something else.

You can treat all TCP/IP data transfer as web data or to be more precise, you can treat data passing through port 80 as web data. ( As port 80 is default http protocol), but technically speaking http server can listen on any available port.

So, my second assumption about your question is that you know which port is being used for web data transfer.

Now you can use various available tools to monitor network traffic. I have used OpenNMS and MRTG based softwares.  Some routers also has inbuilt feature to log data. You can use log files provided by routers to analyse the data.

P.S. You may be just looking for something to monitor your web server traffic on your system, in that case router word in question is confusing. You can use apache logs( if you are using apache) or other web server log file analyser life awstats for this purpose.