My name is Beverley. I've always understood that the name means "beaver meadow" as in "beaver lea", "lea" being the Old English word for meadow.
A beaver meadow, apparently is the rich field that develops from an old beaver lodge that has silted up the river so much that that part becomes a meadow.
In England the name Beverley is used for both sexes. A famous living person is Beverley Goodway, the man who takes all the photos for Page 3 of the Sun newspaper.