What are the criteria for a user background to become login background on user selection?

The image file for the wallpaper must have its permissions set to be world readable. Note however, that only the image file is used on the login screen. It does not use the background colors, gradients, transparency, or scaling mode settings which the user has chosen. Also, I don't think animated or automatically updated wallpapers (such as the "Slideshow" wallpaper selection) work for this.

To make the wallpaper world readable If you go to the file in the file manager, and open its properties, there is a permissions tab. The setting for "Others" in there must be set to at least "Read", and not "None."

The default set of wallpapers in Ubuntu, which is installed in /usr/share/backgrounds are all of course readable by everyone; as are the wallpapers in the gnome-backgrounds package. Custom images in the user's home directory shouldn't be, in general.