“Biewer” is the last name of Mrs. Biewer from Germany. You pronounce it exactly like the animal Beaver.
Here is a short video of Mrs Biewer from Germany, in 2009. She explains how a German Dog Show Judge and vet suggested to name the breed after herself Biewer Yorkshire Terrier 🙂
In 2006 the name was changed into Biewer Terrier. It was the first breed to be established as a purebred through scientific data rather than pedigree data which is customary. The Biewer Terrier was accepted into the AKC in 2014.