Each successive use of rename causes the rename dialog box to move up.

Which version of Windows are you using? 7,10,11, other?

Windows 11.

If you move it down, does it still creep up the next time you open it? Does it continue from where you left it or does it just keep going up no matter what you do?

If you move it, this will continue to happen from the new position.

And just to be clear about what is happening.  Each time the rename dialog box opens it's in a slightly higher position.  Rename enough things and it'll eventually reach the top of the screen.