The bishop Ariadne uses is from what’s called the Staunton chess set (circa. 1850), in which the head of the piece is a designed to look like a stylised Bishop’s mitre. In classical Greek the word mitre (μίτρα) is related to the word mitos (μίτος) which means thread.
Et voila: Ariadne, Mistress of the Labyrinth, and her thread (to go with her symbolic colour, red.)
The other theory that’s just popped into my head is to do with the fact the bishop is considered as useful (if not more) in the endgame of a chess match as a knight- it’s just as powerful certainly, although in a different way- because it can move into a skewer or pin, that is forcing a defending piece to move & expose a more valuable piece behind it to capture.
If you want to pick up the chess analogy & run with it: Ariadne (bishop) is equal to the knight (Arthur? Certainly defender of the King- who is Cobb btw; his name means Lord or Leader in old English) but only she can complete the long move that pins him (or Mal- the Queen?) & forces him (or Mal) to expose the more valuable piece (Mal, or Fischer; the mark, or pawn if you prefer, who perhaps has been promoted by going all the way to the other side of the board/dream?) but that might be reaching
