This is the same question as Save current file with a slightly different name, but for Helm. (The top answer given for the original question doesn't work in Helm, sadly:
next-history-element) doesn't do quite the same thing as in vanilla Emacs.)
The question is: how can I insert the filename of the current buffer into the minibuffer during
write-file. So for example:
C-x C-w (this gets me to a minibuffer prompt with the current directory); in plain Emacs I could type
M-n and this would insert the current filename for me to edit to a slightly different name. What can I do instead in Helm?