Happens on my home system as well (32-bit XP SP3). To be more precise, I need to click outside the rename dialog, but on a lister rather than the desktop background to trigger this.
However, it is probably only a paint issue, since moving the dialog window or even a simple mouseover on the "New name:" field causes a (partial) repaint---the edit box is there, but it doesn't always get repainted. (I have themes enabled, BTW, unlike xbprm.)
It only happens if Desktop Composition is disabled (which is all the time on XP) and only if you click on the lister which opened the Rename window. (Clicking on another window seems fine.)
(Also takes several tries and rapid clicking to reproduce on Win7 while it's triggered easily on XP, but maybe that's just differences in speed or graphics drivers between my real machine and my XP virtual machine.)
As Alan says, it is a painting issue. The control is still there, it's just not being redrawn properly when this happens. Moving the mouse over it or clicking on it should make it redraw, as a workaround for the time being.
A possible fix for this will be in the next beta (not sure when that will be released yet).
Hard to be 100% sure it will fix it in all situations, but I can no longer repro it after a change was made, including in my WinXP VM where it was easy to reproduce before.