This can increase a user clicking the question, and also prevent showing the user irrelevant questions.
But, developing this feature will be harder than hiding individual sites on the sidebar.
This will help prevent unwanted questions from being shown in the HNQ.
I say we should implement it.
You can write a complicated userscript, but the simplest solution is to hide the entire site with a userscript.
Here is a example (taken from here) (set the ignore variable to a comma-seperated list of sites to hide):
// ==UserScript==
// @name Hide SE site from Hot Questions
// @namespace stackexchange
// @description Hide certain Stack Exchange sites from the Hot Network Questions list.
// @include *stackoverflow.com*
// @include *superuser.com*
// @include *serverfault.com*
// @include *askubuntu.com*
// @include *stackapps.com*
// @include *mathoverflow.com*
// @include *.stackexchange.com*
// @version 1
// @grant none
// ==/UserScript==
/*
For list of Stack Exchange sites, look here:
http://stackexchange.com/sites?view=list#traffic
Put your excluded sites here under variable 'ignore'. Example, if you want to ignore Arqade and Mathematica:
var ignore="Arqade,Mathematica";
*/
//This is just an example list, I have nothing against these sites, they just happened to have Hot Questions when this was made
var ignore="Programming Puzzles & Code Golf, TeX - LaTeX";
var questList=document.getElementById("hot-network-questions").getElementsByTagName("li");
var curSite="";
ignore=","+ignore.replace(/, /g, ",");
for(i=0;i<questList.length;i++){
curSite=questList[i].getElementsByTagName("div")[0].title;
if(curSite.indexOf("Stack Exchange")>1) {
curSite=curSite.substring(0, curSite.length - 15);
}
if(ignore.indexOf(","+curSite)>-1) {
questList[i].style.display = "none";
}
}