iPad mini: Download images from webpage preserving file name
Some partial solution's I found since:


Using an alternate browser that supports file downloading. Or mercury browser, which doesn't ask for a file name when saving images, but preserves the original file name and is a full alternative browser with features missing from both safari and chrome such as fullscreen and text zooming (though the latter is served by bookmarklets in safari and chrome). Update: "Documents by Readdle" is another viable option; Most notably it provides a "Save As" style dialogue when downloading files or images, which allows changing the file name right away. 
Chrome download enabler. Sadly, this is a jailbreak tweak, so I didn't try it, but it would give the advantage of having downloads in the same browser that also syncs tabs and bookmarks with the desktop.