A nice photo of the Tiber river... it's a wonderful place isn't it?
It is probable that the name Tiber is pre-Latin, like the Roman name of Tibur (modern Tivoli). A mythic king Tiberinus, ninth in the legendary king-list of Alba Longa, was said to have drowned in the Albula river, which was subsequently renamed the Tiber. The myth may have explained a memory of an earlier pre-Indo-European name for the river, 'white' with sediment. Legend says Rome's founders, the twin brothers Romulus and Remus, were abandoned on its waters, where they were rescued by a she-wolf.
On the other hand, if the Etruscan name, Tiberius (Thefarie), is based on the name of the river, then the pre-Indo-European language would be Etruscan, and the name would be *Thefar. Whether this would mean "white" is unknown. The closest known Etruscan word to it is probably thafna, "cup." In any case it would be surprising if an Etruscan settlement, Ruma, in Etruscan territory, would not be situated on a river with an Etruscan name. This River is said to be the river on which the city of Rome was started (more).