It depends on when your great-grandmother was born, but Briony and Bryony don't seem to have been used as Victorian names at all, so I'm not sure the other posters are right. I found a couple of Brinnies in the England and Wales births, marriages and deaths, one in 1888 and one in 1902. Both of them were born in Wales, where there were several female variants of Brynmor (a male name), like Brynwen and Bryngwyn and Bryneg. Brinley was a popular male name in Wales, too.
Of course none of this may have any connection with your Brinnie!