Elba II was the second planet in the Elba system. It had an atmosphere that was poisonous to humanoids.
The Federation took advantage of the inhospitable environment for the establishment of an asylum for the criminally insane – the Elba II asylum, in operation in the 23rd century. Among others in the asylum was Garth of Izar, a former Starfleet captain.(TOS: "Whom Gods Destroy")
According to the Star Trek Concordance, "the planet is named after the island of Napoleon I's first exile."