The party should not attack him when Byblos is casting Drain at low HP, unless they can deal more damage than he can drain. A Black Mage and Mystic Knight should rely on Fira and other similar abilities to take advantage of its weakness to Fire magic. Using Ifrit in this battle is not recommended; although it deals a lot of damage, Byblos will counter Ifrit's attack by casting Toad and Confuse on a character.
A safe, quick and easy way to win is to use the Blue Magic spell Death Claw to take him down to single digits. Obtaining a Fire Rod from Karnak and using it should also defeat him quickly.
The name likely refers to the Byblos syllabary. [view·edit·purge]Byblos syllabary, also known as the Pseudo-hieroglyphic script, Proto-Byblian, Proto-Byblic, or Byblic, is an undeciphered writing system known from ten inscriptions found in Byblos.