Archive a Hangout

If you don’t want a Hangout to show in your list of Hangouts, you can hide it by archiving it. Archiving doesn’t delete the Hangout or your message history, it simply prevents it from displaying in the Hangouts list. 
To archive a Hangout:
  1. Click the gear icon  at the top of the Hangout window.
  2. Select Archive Hangout.
  3. Click Save
When the person sends you a new message or you send the person a new message, the Hangout will appear in your Hangouts list again. 

 Gmail:

Your saved Hangouts messages are archived in the "Chats" label. Messages from one Hangout may be saved as multiple threads. You can also label, star and trash your Hangouts messages.

When you search for messages in Gmail, matching Hangouts messages will show up along with matching email conversations. All Hangouts are marked with an icon () so you can easily distinguish them from emails.