A standby crew or backup crew was a team or group of individuals assigned to a ship's crew that was able to step in to replace any member of the primary crew when they were incapacitated or leaving their post. They were in slight contrast to their relief, who took over for various reasons, such as shift change.
In 2266, several of the members of the USS Enterprise's phaser crew became incapacitated during a battle with a Romulan Bird-of-Prey, leaving Robert Tomlinson manning the phaser banks alone. Since there was no standby crew available, Lieutenant Stiles volunteered his services. (TOS: "Balance of Terror")
While the USS Enterprise-D was under attack from an Echo Papa 607 while in orbit of Minos, Geordi La Forge, the ship's acting captain, decided to perform a saucer separation, and in preparation he ordered Lieutenant Logan to take command of the saucer section, before requesting the backup crew to report to the main bridge to work with Logan. (TNG: "The Arsenal of Freedom")
During The Doctor's formal proposal to create the Emergency Command Hologram, Captain Kathryn Janeway found the principal of it to be an interesting idea, summarizing it as "a backup captain in case I'm incapacitated and the command structure breaks down." (VOY: "Tinker Tenor Doctor Spy")