As seen in Contacts, Mail, etc the typical behavior in iOS for editing an object (or list) is to have a button "Edit" on the right hand side of the top toolbar. Clicking this button then alters the view to reveal what's editable.
In the iOS files app, to rename a file you click a list item "Rename" (which is not a place or an object, it's an action). This then puts a blinking cursor behind the name of the file, but it doesn't alter the view. I can still interact with whichever elements aren't covered by the keyboard and the name of the file doesn't become apparently more "editable" like a contact's name does in the Contacts app.
It would probably be better to remove the "Rename" list item, add the "Edit" toolbar item, Make sure that the label clearly changes to an entry, and make insensitive (or remove) any widgets that don't have to do with changing the item's state when the "Edit" button is clicked.
Moved "Rename" to "Edit" and placed it in the nav bar. Don't yet have the other actions disabled while in edit mode. Will add that as a subsequent bug.