Gmail – Copy Email from Google Hangouts Chat Window

Click on the Contact to open the previous Hangout you had with them. Click on the Gear and "archive" the hangout.

As of April 2018, It actually is possible to create URL to link to a chat with a specific person. (But who knows when Google will break these given the ongoing incoherence of their messaging applications…)

First, you'll need to know that person's Gaia ID (an internal ID used by Google Hangouts and Google Plus)

• If you're looking for your own Gaia ID, login to your Google account and visit https://plus.google.com/me. This will redirect to a URL of the form https://plus.google.com/1XXXXXXXXXX. The long number at the end of the URL is your Gaia ID. (If you don't have a Google Plus profile, it'll be an error page, but it'll still have the correct Gaia ID.)
• If you're looking for someone else's Gaia ID, search for them on Google Plus. If you can find a URL of the form https://plus.google.com/u/0/1XXXXXXXXXXXX/... associated with their profile, you've got their Gaia ID.
• If you use the reverse-engineered purple-hangouts library to use Hangouts in an instant messenger desktop client, then the "About" or "Info" box for a contact should show you their Gaia ID.

Anyway, once you've got someone's Gaia ID, it's pretty straightforward to create the correct URL to start a Hangout with them. Simply replace 1XXXXXXXXXXXX below with that user's Gaia ID:

https://hangouts.google.com/chat/person/1XXXXXXXXXXXX