In the Connect conversation selector side pannel, some items have Null as the title of the conversation instead of the record number.
The record associated with the Connect Conversation is on a table whose title points to an empty field.
Set the title (sys_ui_title) record for that table to point to the number field or whichever field is desired.
Steps to find the table associated with a conversation and set the Title for that table.
1) Click the conversation in the side panel to open it up in the mini window.
2) Click the View Document button (looks like a square with 3 lines in it) in the chat window header to load the record associated with that conversation.
3) In the URL, you should see the table for the record such as 'sn_hr_core_case', copy this.
4) Navigate to sys_ui_title.list
5) If a record exists for that table, you can simply edit the fields field on the record to be Number.
6) If a record does not exist, create a new one and set the table and fields correctly.
The only documentation we have for the sys_ui_title table is under mobile UI as it also sets titles there.
Navigation by URL explains more on how to find the table in the URL.