To rename/edit a conversation member's name you need to open up this member's Profile view.
There are two ways how you can get there:
Option 1: Open the Members tab in the conversation, click on the person's name and choose Profile from the menu that appears.
Option 2: Alternatively you can click on the member's avatar in the conversation and and choose Profile from the drop-down menu that appears.
In the Profile view, click on the Edit icon (next to the member's name). This will allow you to make changes to the name and rename it to your liking.
When you've made your edits, hit Enter. This change will be visible only to you!