I've a 6.0.10 installation which doesn't remember login.
If I select or leave blank the checkbox, it will always ask me for auth.
I checked cookies, and compared to a working 6.0.10, on the bad installation I get a ZM_TEST bonus cookie. Don't know if this can matter, but that's the only difference I spotted.

I did some search here on the forum, found a couple of threads but were of no help, since the admin settings are correct.

What else can I check?