There is no real consensus, or any idea how that came to be. One theory is that the name Iberia stems from the Phoenician "'ebr" which means "over there," and the use for both the peninsula and the Caucasus was a typical thing in antiquity to have parallels (see Albania, the country and also a kingdom in the Caucasus, for another example).

However in his book Edge of Empires: A History of Georgia (which is where the above is also sourced), Donald Rayfield suggests the term (at least for the Caucasian Iberia) comes from the Speri, a group located in northeast Anatolia that Herodotus called the Sasperi (he also notes it is the origin of the Armenian word for the Georgians (though I cannot corroborate this as I don't know Armenian).