The earliest term the Greeks used was in fact Kaspien thalassan, "Caspian Sea" (Herod. Book 1.202.4; Book 4.40). It was named for a people called the Caspians. According to Herodotus, the Caspians were located in two different tributary districts (nomoi). In one (Herod. 3.92), they were associated with other tribes called Pausicae, Pantimathi, and Daritae. These are otherwise unknown. The other district has the Caspians associated with Sacae. Old Persian inscriptions do not seem to mention them, although some consider the the Persian "land" of Akaufachiya, "table-land" as perhaps "Caspia".
Later Greek sources know of the Caspians as inhabiting the westernmost "circuit" of the Caspian Sea. According to Strabo:
"and in a circuit round the [Caspian] sea after the Hyrcanians one comes to the Amardi, Anariacae, Cadusii, Albani, Caspii, Vitii, and perhaps also other peoples, until one reaches the Scythians;" (Book 11.8.8)
According to this passage, the Caspians were neighbors to the Albani which we know were near or in the Caucasus Mts. Its interesting that Strabo called the Caucasus Mts. the "Caspian Mts". He describes their customes as similar to that of the Scythians and Bactrians, but he never says explicitly that they were Scythians or similar to Scythians, and neither does Herodotus. Their district was called Kaspiane and was in the time of Strabo under Albanian rule:
"To the country of the Albanians belongs also the territory called Caspiane, which was named after the Caspian tribe, as was also the sea;" (Book 11.4.5)
If there was an Avestan equivalent to 'Caspian sea' it may have been the term Vouru-kasha, "broad inlets" or "wide bays" and described as vraya, "sea", hence "vraya vouru kasha".
http://www.cais-soas.com/CAIS/Religions/iranian/Zarathushtri an/avestan_geography.htm
Two rivers, the Rangha and Vanghuvi Daitya flow from it. The Vanghuvi Daitya was otherwise known as the Oxus, and we know that a branch of the Oxus flowed from the Caspian. It may well be that the origin of the name "Caspian" may not have lain with the name of the tribe but as a descriptive, from kasha of Avestan sources.