Sincnoel, you raise an interesting question. Do you mean that the published solution could be either word, or that either word would be acceptable. Or even that both words would appear in print (I've never seen that).
Going by 1940s primary school spelling it was briar. I have a theory that alternate spellings such as this (brier)become acceptable after a period of common usage ie: bad spelling ,lol.
"Brier Pipe" . . . . WEB 1030 . . . . . UK 33
"Briar Pipe" . . . . WEB 50900 . . . . UK 1720
(For those of you who are not sure when to use quote marks this is the way to search for a group of words or numbers as one block of characters, ie with no separators.)
Thanks to you all for your answers and views. I'm going to defy all (well, nearly all) logic, and go for Briar - simply because it looks right to my eye.