WHen you open manage | Database and rename a field, all references to that field in scripts, calculations and relationships automatically update.
The layout text on a layout may or may not update to match. If you have not edited the layout text after it was automatically added to the layout when you added the field, it will update. If you changed the text added--often field names have "internal use only" naming conventions that have to be edited to avoid confusing users--the layout text will not update.
Also be careful of any calculations in fields or script steps that refer to your field by name by using text in quotes such as in a set field by name step. That quoted text will not update when you change the field name.