There was a great deal of discussion in the sci.math newsgroup about a decade
ago. See the threads
[Beal's Conjecture][1] and [Against the term "Beal Conjecture"][2].
As with most sci.math discussions, they generated more heat than light.

  [1]: http://groups.google.co.uk/group/sci.math/browse_thread/thread/31cd9a6140a743f5/f3d6fc0644a98481?hl=en&q=sci.math+beal%2527s+conjecture&lnk=ol&
  [2]: http://groups.google.co.uk/group/sci.math/browse_thread/thread/d8d2ad7d61a796e3/1755be7ffc2bfd5a?hl=en&ie=UTF-8&q=sci.math+beal%2527s+conjecture#1755be7ffc2bfd5a