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The issue mostly arises in domain separated instances. The chat window is showing sys_id of the user instead of the name because of a duplicate record of the user in the live_profile table with sys_id in the name field. 

Kindly follow the below steps to find the invalid record of the affected user. 

1) Go to "sys_user" and get the sys_id of the affected user 

2) Go to the live_profile table

3) Use a filter, document = <affected user's sys_id>

4) This returns multiple profiles for the same user, in that find the profile which has a sys_id in the Name field.

5) Update the record with the User name in the place of sys_id.

Note: To find all the culprits records in the instance and update it with a valid user name, write a background script on Live_profile table and check the name contains 32 char alphanumeric value and if it returns a sys_id then do a glide Record on the sys_user table with the sys_id and get the name of the user. Replace the sys_id with the User name and do an Update operation on the live_profile table.

Script: Attachment "Background_script"

To fix this issue permanently, write a before insert/update business rule on the live_profile table and validate the user name and correct it if needed. 

Script: Attachment "Business_rule"