Following on from the post on Ajax in Taxila, we should note that Pliny the Elder (NH VIII v 11-12) tells us about some other named elephants.
King Antiochus (unnumbered, sadly) had at least two, one called Ajax, the other Patroclus. Pliny tells the story of reluctance, rivalry, and reward, on an occasion of fording a river. They were celebrities of their time: duo... celebres etiam cognominibus fuisse.
Cato the Elder (apud Pliny loc.cit.) excluded the names of generals from his Annals, but not that of Syros, 'the Syrian' altero dente mutilato.
Plutarch name in the Life of Pyrrhus 33.4 [.5 in the Teubner] a battle-elephant named Νίκων 'Victor'.