writing a trigger in order to change Table Name automatically
Sys.Trigger_event_types will have all the events that can be fired.

On checking, it has an event type called RENAME. You could use that write a DDL trigger, though this may not be ideal but serves your purpose.

alter trigger  tst
on database
for
RENAME 
as 
begin

declare @oldtblname sysname,@newtablename sysname
  SELECT 
       @oldtblname= EVENTDATA().value  
        ('(/EVENT_INSTANCE/Parameters/Param)[1]','nvarchar(max)') ;

          SELECT 
       @newtablename= EVENTDATA().value  
        ('(/EVENT_INSTANCE/Parameters/Param)[2]','nvarchar(max)') ;

exec otherdbname.dbo.sp_rename @oldtblname,@newtablename;

end


There are some catches, since this is database level scoped event, this will fire for all rename events, you may want to check event type and if it is only &lt;ObjectType&gt;TABLE&lt;/ObjectType&gt;, then you may wish to continue.

Note:
I have  SQLServer 2012 and 2005 machines, this RENAME didn't worked on 2005 but worked on 2012. Not sure about 2008.