How to rename mailbox in latest IPhone software.

If you mean rename the mail account, go to Settings/Accounts & Passwords, tap on the mail account you wish to rename on the right side, then under “Description” edit to your choice.

This is an old article so locations are different (follow above for iOS12), but the final part is the same so should give you the idea.

http://osxdaily.com/2014/04/02/rename-mail-account-ios/

It might have something to do with type of account, mines still pop and I see this.

If I tap on "Pipex" next to Description I can then edit the text (but there's no edit button).

Glad you sorted, however you managed it, FWIW the Mail Account Screen is the same whether iPhone or iPad.