Hesiod gives the name as Clymene in his Theogony (359) but Apollodorus (1.8) gives instead the name Asia as does Lycophron (1411). It is possible that the name Asia became preferred over Hesiod's Clymene to avoid confusion with what must be a different Oceanid named Clymene who was mother of Phaethon by Helios in some accounts.
Herodotus (4.45.1) records the tradition that the continent Asia was named after Asia whom he calls wife of Prometheus rather than mother of Prometheus, perhaps here a simple error rather than genuine variant tradition. Both Acusilaus and Aeschylus in his Prometheus Bound call Prometheus' wife Hesione.
"According to the Greeks in general, Libya was so called after a certain Libya, a native woman, and Asia after the wife of Prometheus. The Lydians, however, put in a claim to the latter name, which, they declare, was not derived from Asia the wife of Prometheus, but from Asies, the son of Cotys, and grandson of Manes, who also gave name to the tribe Asias at Sardis." Herodotus
See Prometheus for variant accounts in which Prometheus' mother was Themis or Hera.