A glance at the word 'bryony', from which the name comes, will show that in UK English at least it is pronounced brie o ni, with the first syllable having the same sound as in 'eye'. If a good American dictionary shows something different, then that is the American pronunciation.
Either way, Bryony
has got nothing whatsoever to do with Brian
. Look them both up, and you'll see. The Briony
spelling is used for the plant, but less frequently than the Bryony
version, which goes right back to ancient Greek via Old English. So, both spellings are accepted, but your pronunciation would raise eyebrows among speakers of British English.