In my time having it as my first name, I have found a keen interest in it. True it comes from the deity Tanit (5th Century BC) who was a sky goddess and there is still a temple foundation to her in Tunisia today. Her consort was Ba'al-Hammon, the God of the Sky her accomplice who is now often linked to darker magics. She has the symbol of Tanit shown as a woman stick figure with upturned arms she is associated with the moon, stars, fertility and yes, I have heard serpents as a symbol. The Palm Tree is also connected as a kind of desert version of the tree of life (links to the fertility to produce life in a barren landscape) and her connection to a mother earth figure leads to the symbol of the snake. Also associated with Ashtart (Astarte) and Athirat her symbols include doves, grapes, pomegranate, crescent moon and in along with Ashtart the Lion. She was assimilated into the cultures of the roman invaders, her deity was borg style mutated and hence the success of the Romans as they pulled in the religions of the cultures they trampled all over. She is then associated with Juno and given the title Dea Caelestis, "the Heavenly Goddess", or Virgo Caelestis, "the Heavenly Virgin". Other things I read were that they found a lot of child bones in an area of her temple ruins. Now the academics of the world are thankfully split on the meaning but I know what side I am on. One half believe that the Carthaginians sacrificed the young to their lady and god but another group said that there are no child burials only the older were buried in graves and I think due to high child mortality rates that they took the young who died to her for her to take care of in the afterlife, a kind of heavenly stepmom. Her legacy was spread through the soldiers who then traveled the world and she reached the culture of Spain, Malta and Sardinia as the symbol of hers is on the temple of the acropolis in Selinus in Sicily and although I haven't seen the pictures, a shrine in Rome on the north side of the Capitoline Hill. She is also linked/associated with Aphrodite, Demeter, and Artemis by the Greeks. The Romans also linked Her with the Magna Mater, the Great Mother, Rhea or Kybele. (which I found even more amazing because for no reason they know of my parents don't know how they came up with the name 2 days before, but my middle name is Rea which is just another way to spell Rhea. Both my parents were Leo starsigns, the Lions) Rhea (just for fun) was the mother of the gods of Olympus and bore Hestia, Hades, Demeter, Poseidon, Hera, and Zeus and in that order her hubby swallowed them fearing a prophecy that he would be dethroned by one of his kids. Rhea saved and hid Zeus with the help of her own mother Gaia who was able to later return to face Cronos and have him basically barf his siblings back up. She herself had no real following and her depictions are 4th century BC but interestingly there are two lions pulling her chariot, or she is flanked by two lions. Check out the Lion Gate of the Mycenae acropolis.
Additionally, the snake link. I have seen the symbol of the snake in connection with her actually depicted like the symbol we recognize for medicine, the caduceus. The palm tree of life or sometimes a staff and two intertwining snakes. (adding yet more weirdness to my connection that both parents were 'lions' or Leo signs but one was an ambulance driver, the other a nurse) Simply google the symbol of Tanit and you will see her signs and where you can see them around the world. Haven't found any museums in the UK or nearby with anything yet.
In ancient egalitarian cultures, feminine energy was revered and the serpent represented the life force. I am not sure how the name sounds, but I love the meaning.
In Spanish it's written Tanit. In the Phoenician times in the island of Ibiza (were I live), the goddess Tanit was the patron saint of the island. Just a little bit of history there. :)