The name Baz is a boy's name meaning "regal; falcon". Baz and is often added to lists like Vintage Nicknames for Boys and discussed in our forums with posts like "Namebank CAF -- Charles Dickens Edition (7 or more children.)".
From the experts:
Curiously, Australian-born Moulin Rouge director Luhrmann was born neither Sebastian nor Basil, but had the name Mark Anthony on his birth certificate; his nickname arose from his supposed resemblance to a British TV fox puppet named Basil Brush.