Answer
The named chat stays open, and your next message goes to that same chat. Cancelling leaves the new chat unsaved.
Saving the name adds the chat to the AI tab and makes it eligible for chat search, even without a user-written message. Selecting its title now opens Chat Options, including Search Messages, Media & Files, Advanced Options and Delete Chat.
The rename adds an activity message such as Renamed chat to “Dog” and updates the chat-list preview. It does not request an AI response. Renaming an existing People chat records the same kind of activity.