I know, this is extremely annoying.
You can try this on Chrome. This worked for me (with a pinned tab), in February 2017.
Create folder somewhere on your hard drive and create these two files in it:
manifest.json
{
"manifest_version": 2,
"name": "Pinned Tab Flashing",
"version": "1.0",
"description": "Disable pinned tabs flashing",
"content_scripts": [{
"matches": ["https://www.messenger.com/*"],
"js": ["content.js"]
}]
}
content.js
function titleModified() {
var text = document.getElementsByTagName('title')[0].text;
if (text != 'Messenger') {
document.getElementsByTagName('title')[0].text = 'Messenger';
}
}
window.onload = function() {
var titleEl = document.getElementsByTagName("title")[0];
var docEl = document.documentElement;
if (docEl && docEl.addEventListener) {
docEl.addEventListener("DOMSubtreeModified", function(evt) {
var t = evt.target;
if (t === titleEl || (t.parentNode && t.parentNode === titleEl)) {
titleModified();
}
}, false);
} else {
document.onpropertychange = function() {
if (window.event.propertyName == "title") {
titleModified();
}
};
}
};
Navigate to chrome://extensions/, click on Load unpacked extension
and choose the folder you created. Then refresh your Messenger page.