In a third-order tensor I guess the first and second index would be called row and column respectively but is there a name for the third index?
A common name after row and column is tube, according to Google. More generally, one speaks of fibers of a tensor, enumerating them by modes. The illustration below is from the book Nonnegative Matrix and Tensor Factorizations by Andrzej Cichocki, Rafal Zdunek, Anh Huy Phan, and Shun-ichi Amari, published by John Wiley & Sons in 2009 (I surely hope reproducing one illustration is fair use).
(And +1 to the question; it received one of the more unreasonable downvotes I've seen around here.)