How do I rename a book mark on my iphone?

You have to be viewing the Bookmarks screen. Click on the Bookmarks Icon at the bottom of your screen to view a list of your Bookmarks:

Tap on, and hold the tap, until you see various options, one of which is "Edit:"

What happens after you "tap and hold?" In order to see the entire display be sure to "scroll up."

You are welcome.