With its meaning of "vine," this great baby name, Greek in origin, also means "to sprout" or "to grow luxuriantly," and it can be pronounced bree-ON-ee or BRY-an-ee, according to preference.
Famous literary Brionys include Briony Tallis of Ian McEwan's novel Atonement, who was played by Saoirse Ronan in the film adaptation of the same name. Then there's the late, great ballerina Bryony Brind, who spelled her name with a Y, actress Briony Glassco, and 1970s cult film actress Briony Behets.
Baby Center charts indicate that Briony was very rarely used in the U.S., circa 1980-2008, but in 2012 it started to climb (like ivy or another species of vine, if you will) a bit higher. It is, however, quite popular overseas, to the tune of being one of the top 100 female names in Britain. Which makes it unlikely to belong to any other child on the (American) block.