Tanith does mean "serpent lady" or "snake goddess." She was an important goddess to the Phoenicians, ruling love, fertility, and the moon like so many of her counterparts in the ancient world. Sometimes it is also said to mean "sky goddess" or "celestial goddess," but these are just as likely to be epithets or aspects. She was the chief goddess of Carthage, consort to Ba'al.
Phoenicia.org lists this inscription as found on a tablet from Carthage: "To the lady Tanith, and to our master, the lord Baal-Hammon; the offerer is Abd-Melkarth, the Suffes, son of Abd-Melkarth, son of Hanno." There are many other documented inscriptions to her on tablets and votives.
The name Abd-Tanith, "servant of Tanith," is commonly found in inscriptions from the region.
Other forms she was known by include: Tanata and Anaitis. It is related to the Greek name Tanis, as well as the Punic form Tent. It may be related to the Egyptian Tanetu.
Information from "The History of Phoenicia", chap. 16 : "Tanith is invariable placed before Baal, as though superior to him, and can be no other than the celestial goddess (Dea cœlestis), whose temple in the Roman Carthage was so celebrated.90 The Greeks regarded her as equivalent to their Artemis;91 the Romans made her Diana, or Juno, or Venus.92 Practically she must at Carthage have taken the place of Ashtoreth. Apuleius describes her as having a lunar character, like Ashtoreth, and calls her "the parent of all things, the mistress of the elements, the initial offspring of the ages, the highest of the deities, the queen of the Manes, the first of the celestials, the single representative of all the gods and goddesses, the one divinity whom all the world worships in many shapes, with varied rites, and under a multitude of names."93 He says that she was represented as riding upon a lion, and it is probably her form which appears upon some of the later coins of Carthage, as well as upon a certain number of gems.94 The origin of the name is uncertain. Gesenius would connect it at once with the Egyptian Neith (Nit), and with the Syrian Anaïtis or Tanaïtis;95 but the double identification is scarcely tenable, since Anaïtis was, in Egypt, not Neith, but Anta.96 The subject is very obscure, and requires further investigation."
It is uncommon as a modern given name, but there are a few notable namesakes, such as TanithLee, the author.