I have just started tracing my family tree and so haven't got very far yet. However, I'm very interested in your line of research because there is an oral tradition in my family that we are related to Sir Thomas More, and its true to say that my grandmother (!) did look very like some of the portraits of him!
Her maiden name was in fact Moor, but I understand that Mooor, More and Moore may be interchangeable. Her father's name was Cresacre Moor - i don't know if that is a throw back or a pretnsion on thier part. Another relative was called Margaret Roper Moor - no doubt after Sir Thomas' favourite daughter.
So far I have been busy trying to trace Sir Thomas' male descendents as best I can to see if there is apossibility of a line of descent. I know Sir Thomas had only one son, Jon who had only one son, Thomas II (?), who in turn had three sons, Henry, John and Chrsitopher Cresacre. I have only been able to trace C.C. and I believe he had a son called Thomas, who had a son called Basil , who had a son called Christopher Cresacre. Which still only takes us to around 1693. So its slow progress.
If you have any leads on Henry or Johns descendents could you let me know, and in return I'll drop you a line if I manage to crawl up the family tree any further from my end!