In most parts of FileMaker, you can open Manage | Database | Fields, find the field and change it's name and it's reference in other areas such as calculation fields, scripts and layouts will update automatically. The exceptions to this are where you have exprssions such as SetFieldByName where you refer to the field in literal quotes. If you change a field name, Get (ActiveFieldName) will automatically return the new field name, so that may work for you or not depending on what you do with that field name in the rest of your expression.
One trick when passing a field name in a script parameter is to use GetFieldName instead of putting the actual field name in quotes. If your field is named "FirstName", you'd put GetFieldName ( Table::FirstName ) instead of "FirstName" as your parameter. Then changing the field name in Manage | Database doesn't require an update of your script parameter expression.
If you have used setFieldByName or other such expressions where you've identified field names with literal text, you might want to use FileMaker Advanced to generate a database design report that you can the search for instances of a given field name to make sure that you don't miss one that should be updated when you change the field name.