To set up the bookmarklet, go to https://www.squarefree.com/bookmarklets/forms.html#remember_password
and copy link address to clipboard (there are other sites with similar bookmarklets; this is one that works for me). Create a new bookmark with any name you like, and paste from the clipboard to the bookmark address field.
You will need to use the bookmarklet each time you visit a "problem" site, so you may decide this is more trouble than it's worth. To use the bookmarket, go to the site with the username/password field, then bring up Opera's bookmarks page and tap the bookmark. Opera should then return to the username/password page. You must then activate the username field by tapping it. If nothing has previously been saved, type username/password as usual and Opera should offer to save. If the password has previously been saved, tapping the username field should cause usename/password to be filled in (if there is more than one username associated with the site, tapping the username field should bring up a selection box).
An alternative to bringing up Opera's bookmarks page and tapping the bookmark, is to start typing the bookmark name in the URL/search field. With default Opera settings, the bookmarklet should eventually be shown as a search suggestion. Tapping that suggestion will run the bookmarklet.
I see on Slashdot someone has created a Chrome extension to put that button back ...
No idea if it would work in Opera (with the Chrome Store extension obviously).
@operes Size of the image? No, sorry. I know people have asked for that option (so it should already exist in the desktop Feature Requests forum - add your vote to an existing thread), but it does not exist yet.