The sci.math discussions linked to above suggest that Andrew Granville suggested the problem in 1992 and that it was discussed as early as 1985.
I have in my notes:
T-Z predates Beal; see Frits Beukers, "The Diophantine equation $Ax^p+By^q=Cz^r$", Duke Math. J. 91:1 (1998), pp. 61-88.
This kind of informal documentation may be the best available, unfortunately.