Heyyo,
Ok, I'll give you some examples of the crap I'm trying to make sound easy cause I bet it's confused ya.. I'm not a very clear speaker, lol.
Let's say you wanna add "ThE_MarD's CS1.6 LS" to your favs. You know the server is running, you add the server to your favs and then refresh, and it pops up in your favs with the player count, map, and your ping. It's now saved permanently to your favorites. If the server goes down and you refresh? it'll still be there but say something like "<not responding>"
Now, let's say you try to add my LS but sadly, I diddn't feel like playing CS1.6 and the listen server is down. You try to add it to your favs, you refresh, and it's not there. I turn the server on, it's still not showing up.
now, in the first case the server will stay there cause it was working when you added it to your favs. The 2nd time it failed cause the server was down.
So yeah, just make sure when you add any servers to your favorites that it's online. One good way is to first join the server, hit ESC, click "find servers" and it should be listed in the "history" tab, and then add it from there. Bam. Done deal.