When a customer is chatting with an agent, by default their extension username will appear as configured in the PBX directory. The name can be changed using the steps below.
If the agent is using Agent Dashboard for chat
As a reminder, the Agent Dashboard is an application installed on the agent's PC. If you do not have an application installed on each agent's PC, you are likely using Contact Center Agent Client which is detailed lower in the article
Once the agent is signed into their Dashboard, right-click on the Chronicall icon in the system tray:
The icon may be hidden inside the system tray. Simply click the Upward Arrow to display hidden applications
Note: the icon applicable for Avaya Call Reporting is 
Once you've right-clicked the icon, select Settings ► Change Chat Nickname
If the agent is using Contact Center Agent Client (CCAC) - Recommended
As a reminder, Contact Center Agent Client (CCAC) is a browser-based application and does not require software to be installed on the agent's PC.
This needs to be adjusted by a Manager or Administrator in the HTML client of the software.
To accomplish this, log into the software using your manager credentials at http://<ipaddress>:9080/web
Select the Realtime icon in the left pane, then Supervisor View
Find the agent you need to modify and select the ellipsis menu button in their pane. Select Edit Profile.
Add the agent's nickname and save your changes.
Note: You may also change the agent's image in the profile menu by clicking on the avatar image.