ἡ δὲ Πυθία τότε πρῶτον Ἡρακλέα αὐτὸν προσηγόρευσε: τὸ δὲ πρώην Ἀλκείδης προσηγορεύετο.In this, Hercules was named after his grandfather Ἀλκαῖος, transliterated as Alcaeus (Alkaios). Another version has him named Alcaeus, again, after his paternal grandfather.
"The Pythian priestess then first called him Hercules, for hitherto he was called Alcides."
It was the priestess at the Delphic Oracle who gave him the name Hercules, only she really didn't. Hercules was a Greek hero, so she would have given him the name Ἡρακλῆς, which we can transliterate as Herakles or Heracles. Hercules is the Roman version of the Etruscan version of the Greek name (Hercle).
In the modern western world, Hercules is the usual name applied to the hero.