According to the Social Security Administration, this name is growing in popularity for girls in the United States. I think it’s a bad idea. I don’t know about the rest of the US, but in Ohio, Kentucky, and West Virginia, “Briar” is a slur term for poor rural Appalachian people. So if you wouldn’t name your little girl “Hillbilly,” don’t name her “Briar,” either.
If it’s one of your family surnames… well, make it a middle name, that’s all I can say. Alternately, call the kid Briar-rose, like Sleeping Beauty. (But in that case, why not just call her Rose?)
It’s also not a good name, because it associates the kid with the bad effects of the Fall cursing the plants of the earth. You know, “and thorns infest the ground”? Briars are also associated biblically with the ‘crackling’ sound of fools talking, with abandoned settlements and fields, and with all sorts of dire prophecies against the wicked and the pagan nations being burned up. The only positive association is the Crown of Thorns, and that’s an awfully sad name for a kid; or the “lily among the thorns,” which is a comparison of the Beloved to other women. And your kid would be the “other women,” not the Beloved.
If you want a Bri- or Bree- name, there are tons of those.
That said, there is a St. Spinella (“thorn”), who was martyred in Rome with St. Felix and her seven brothers (feast: June 27), and a Bl. Spinela who was a Cistercian nun in Arouca, Portugal (Nov. 1).
I suppose you could go with Bruyere, the French version, if you really really wanted to inflict this name on your kid. It means “heather” and “moorland” as well as “briar,” so at least your munchkin will have some positive associations.