AdminCP >> Appearance >> Style properties >> Messages >> User >> Message user info elements
Specifics may depend on your custom style but look for the group called Messages. Click on that and you should see various options for messages and for the Message user info block. One of those should be Message user info elements and there you'll see an option to include Custom fields as well as the standard fields. Use the checkboxes to select the ones you want to display.
You can't select which custom fields, though. It's all or none.