Was it from the Carthaginian Barca family or a native Iberian name?
it's always worth checking an etymology dictionary. according to etymonline:
Barcelona city in Spain, said to have been named for Carthaginian general Hamlicar Barca, who is supposed to have founded it 3c. B.C.E.
Sorry, not linguist nor etymologist here, just a native from the city ... :)
The spanish wikipedia have all the possible origens I know of (http://es.wikipedia.org/wiki/Barcelona)
The only true fact is that the first known name is: "Colonia Iulia Augusta Paterna Faventia Barcino". "Barcino" comes from "Barkeno", the name of an Iberian settlement there.
A phoenician coin with an iberian inscription "Barkeno" supports the theory of it being from Phoenician origin, but is not conclusive.
There are two myths on the foundation of Barcelona and its name:
Founded by Almicar Barca or his son Annibal Barca, depending on the version.
Founded by Hercules and the Argonauts. One of the boats got lost in a storm after the 4th job, and landed near the current Barcelona. This boat whas the 9th boat, the "Barca nona". Seeing it was a nice place, founded a city with that name.