I've added the option to add to faves at the all single pages and made the special page with the favorite posts.
I do not use user registration so it should work with cookies.
The problem is that I could not find a way how to force it work with cashe plugin. Is it any way not to exclude the single page from cashing ? It is very important since the blog is very busy. But not then any user press "add to faves" and reload the page - he/she again see "add to faves".