Each Garth One-Eye keeps track of card names chosen for itself, but not for other Garth One-Eyes. (Garths One-Eye?) If Garth One-Eye leaves the battlefield and returns, or you control another copy of it, it won't remember the choices made during earlier activations. However, if another player gains control of Garth One-Eye, it will remember choices made for it.
Official card text for each card Garth One-Eye references can be found using the Gatherer card database at Gatherer.Wizards.com.
As Garth One-Eye's ability resolves, you must choose one of the listed card names. You may choose not to cast the copy created, but you won't be able choose that card name again later.
Magic: The Gathering · ™ & © Wizards of the Coast · Illus. Micah Epstein
Card data via Scryfall. Rarix is fan-made; not endorsed by the rights holders.
If all choices have been named before, Garth's ability won't do anything when it resolves.
You must pay the costs for the chosen card, but you cast the copy as the ability resolves, which may not be a time that you could normally cast that card.
Resolving copies of permanent spells become tokens as they enter the battlefield. These tokens are not considered “created” when this happens.