I do not know how one renames a text box nor why. I see that you are talking about the graphic "text box" rather than a formfield.

00 deleteme 1.png

What problem is this giving you? I am guessing in a vba procedure. Each new iteration will be in a different section, so you could loop through sections as well. I do not understand the process by which Word adds numbers to identify the native text boxes.

It could be that if you added your text box to the text boxes building blocks and added it from that, it would simply show up as Text Box #. I do not know.

Hope this helps. Perhaps someone else will have some ideas.