Is there a known issue with HTTPS and the Android browser?
I don't think there is any widely known issue with this. I am able to sign in to StackExchange automatically on my Nexus S. I looked around at the Android Issue Tracker and while there are a lot of results for "browser https", none in the first 20 or so seemed pertinent.

It might just be a problem local to your phone. I'd try resetting it.