I like
Bryony, because I like ys, but
Briony is fine, too. I like it pronounced either BREE-oh-nee (as you do) or BRYE-oh-nee (long i*), but the latter is the one I have fixed in my head.
* Is there a standard guide on this website for pronouncing vowels besides IPA? I'll have to look...
*****
And last of all an Admiral came,
A terrible man with a terrible name,—
A name which you all know by sight very well;
But which no one can speak, and no one can spell.
Robert Southey, "The March to Moscow"