Is there a known issue with HTTPS and the Android browser? I have a Galaxy S2 running 2.3.3, all up to date and not rooted.  On several sites I find that logging in is impossible due to requests sent on HTTPS not returning a response.  

I've not tested the issue exhaustively, but the most notable failures for me are my Natwest online banking and signing in to Stack Exchange sites with my Google ID.  In both these cases the response from the server never appears, in the case of Stack sites this happens when clicking the 'Log in with Google' button, I never even get a chance to enter credentials.

I've resorted to using Opera Mobile for now, and since this works fine it seems to be an issue localised to the android default browser.  Is there a way of fixing it, or should I just choose a different browser and set that as default instead?
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.