My setup is Mac OSX El Capitan v 10.11.6 with Safari v 11.0 (11604.1.38.1.7).
The website in question appears to be properly set up in Safari with regard to cookies, and Safari > Preferences > Privacy is set to 'Allow cookies from websites that I visit'.
I have been successfully using Cookie for a long time and the website in question has been accessed very regularly without any problem.
The number of Favourite websites has remained around the figure currently showing for some time.
I am not aware of removing the website in question but it has disappeared from my list of Favourites. The problem came to light about a month ago when I experienced having to Log In every time I moved between areas of the website.
Normally when visiting a website Cookie includes it within the listing giving the user an opportunity to mark it as a Favourite.
In this case whilst the website is being used the listing fails to show that it is present and so it is not possible to mark it as a Favourite.
Being 'Logged Out' every time the user browses another page is frustrating.
However after one re-log on the website details appeared in Cookie very briefly but I was not quick enough to be able to checkmark the Favourites box.
Any help you can give will be greatly appreciated.
my apologies, some users lost their favourites in the upgrade to 5.6 (it looks as though this affected you)
what you need to do, is disable any timers you may have running.
then visit the sites you need to favourite, so you can reselect them.
once you have all your sites reconfigured, you can re-enable any timers you had.