"TheOperator" appears to be a nickname, based on her functions.

From the Phase 1 MaydayTextOrganized:

From the PhaseEmailsFromLadybee from TheSleepingPrincess:

The Operator's memory fragments show that her real name is Melissa.

In one LinksOnFunStuffPageFinalSolution, TheCastaway calls her "Melissa":

In both of the Operator's PhaseShipboardMemories, CaptGreene calls her "Melissa":

Finally, in the PhaseEmailsFromLadybee from TheSleepingPrincess, McKaskill calls her "Operator" and "Op" and "Melissa":