First check to make sure this is what you are doing:
You have a field named "xyz" and you want to change its name to "abc", correct?
Select Manage | Database | Fields and select the table that contains your field. (Other tables may have a different field with the same name.)
Select the field by single clicking it and enter a new field name. Click change to change the field name. Now table view layouts should show the new field name. Other layouts may or may not show the new field name. To change the displayed field name on a layout where it didn't change, enter layout mode and use the text tool to edit the text.
(In filemaker, you can use any text you want as the label for a field, if the text matches the actual field name, it should update when the field name is changed. If it is different in any way, it will not change and you'll have to edit the layout to change it.)