According to this site, and from what I understand, the name is correctly pronounced BRYE-uh-nee.
It could be confusing, though, since this site says it's prn. "BRIE-u-nee" and brie, the cheese, is prn. like BREE. I believe
Bryony and
Briony are both pronounced like a long "i" sound, though. But I could be wrong. :b