The useful thing to do would be to display a notification (that's not hidden) saying someone wants to get in touch, which I can then accept or reject. Like, oh, how MSN Messenger or Yahoo Messenger used to handle this back around 2004 when instant messengers were actually user-friendly.
Possibly the answer is to use something else. WhatsApp seems to be the current cool one, but is terrible in its own way because it only works on your mobile (and yes, I know WhatsApp Web exists - except that only works if your phone is on and connected to the internet, because it's basically a remote control for your phone. And WhatsApp Desktop is a remote control for WhatsApp Web because why do something sensible when you can do that instead). Skype is basically MSN Messenger with all the features removed, the current Yahoo Messenger has likewise been stripped down to the bare bones and then some more (it doesn't even do online/offline status!)... and I'm not sure if any of the other classic ones (ICQ, AIM) still exist. Or have any features left.
Last-minute update: just I was about to post this, the tab for the Facebook website flashed a "(1)" in the title showing that there was some sort of update. It then completely failed to open the chat window, highlight the messenger icon, or do anything at all to show me that I'd just received a message from a FB friend. Because doing any of those would be a useful thing to do, and FB seems determined to not do anything useful. Sigh.