Sure, you can make the names change for each and every field in the entire game, I know what you meant.
No, that isn't how it works - the names can be set from text and then they are ok for the entirety of the game. You are setting the name from text using an operation that was only used originally in the debug menu. You can choose when to use it. Just before getting Cait and Vincent, you'd use it.
@Tsunamix
The default names used in the menu screen are held in the executable (as opposed to the names displayed BEFORE the menu screen, which are held in the kernel (the only four used are EX-SOLDIER, Red XIII [at least I think so], Cloud [using Cait's slot] and Sephiroth [Using Vincent's slot] ). But that won't stop people being able to change the names if you allow the naming screen.
I suppose with a bit of hex editing, you could disable the selection cursor and force them to press OK to every name from that menu. Or you can do what I suggested and remove the menu entirely and use the operation that sets the name from text.
So to sum up:
Choice 1: Allow naming screen but hex edit so they are forced to press OK to the default name (held in the executable - These can be renamed by touphScript too)
Choice 2: Force the name from field text itself (just before you gain the character) - and remove the Naming menu entirely (this means removing the naming screen operation in the relevant field and adding the operation I directed you to in the above post).