FavOrg from PcWorld creates a way to save them. * if it's not available, just check Google for places that have it.
Run FavOrg occasionally to recover the favicons. The favorites list then shows the favicon until site is removed.
Yes it’s a trick, but clients do want it. They see favicon.ico in the log files; ask what it means…..
When given a choice, to have the default IE one show, or theirs,(clients); they ALWAYS chose theirs, even if it has an “ephemeral” life.
People do not like the idea of a file being requested and not found. If there is a file to be pulled up, they want it. Hence, the preference for a favicon, no matter how short lived.