When the new computer just install the Oracle client and the BCB6 (BDE), and BDE pop up an error message “Invalid Filename” when change the server. Here is a simple check point before dive into the Oracle configuration or BDE configuration.
Click the Object in the BDE menu, and point the existing SqlPlus execute file, and try it again.
If got the error message still, maybe try to add the system variable ORACLE_HOME, as this post, and here is the reference to edit the registry.