Fenicia is a kind of place name that means both "phoenix" and "Phoenicia," as in the ancient kingdom. Phoenicia was a Semitic culture in the major Canaanite port towns (Syria, Lebanon, Israel specifically), and from them we developed most modern alphabets. Phoenicia's meaning was not strict - it meant "[land of] purple (dye)," "blood red," "red-dyed wool," or possibly "phoenix." If Fenicia is equivalent to Phoenicia, then Fenicia means purple or red dye as well. Being such and old name, it's hard to tell how relevant the word phoenix is here. The word purple may have the same origins as Phoenicia. Feniccia (fen-EE-chee-uh/fen-EE-cha) is an Italian surname, and Fenica (fen-EET-sa) is an alternate spelling. There are supposedly about 56 people in the U.S. named Fenicia. Fenicia can also be found as an Italian place name.
This name was used by Giambattista Basile in Il Pentamerone, his collection of fairy tales. The story in particular is The Enchanted Doe, in which a king who wants children tries to persuade the gods to grant his wish by giving all he has (money, room and board) to any beggar who asks. Realizing after a while that he is going broke and still has no child, King Giannone shuts his doors and stops being charitable. One day another beggar comes to him and says he can grant Giannone's wish, and Giannone says if he's telling the truth he will give over half his kingdom. Soon twin sons are born, Canneloro and Fonzo, and they are extremely close to each other. The queen grows jealous and tries to kill Canneloro, the son she doesn't favor, but doesn't succeed. Taking it as a sign he is no longer welcome, that son leaves, giving his brother and fountain and myrtle tree by which he can watch his brother's life - if the fountain is running clear and steady, Canneloro is healthy, and if the myrtle tree withers, Fonzo will know his twin has died. Canneloro travels, has adventures, and wins a tournament in the Kingdom of Clearwater, where he is given the Princess Fenicia as a reward. Months later, a shape-shifting ogre in the form of a doe lures Canneloro into the heart of the forest and brings on a fierce snowstorm. He takes cover in a cave, and the doe requests to take shelter with him. He promises not to hurt her (him, in fact) and when Canneloro is completely defenseless the ogre turns back to his real self and locks Canneloro up in a pit to eat later. Meanwhile, Fonzo is watching the fountain and myrtle tree, realizes his brother is in trouble, and goes to find him. He comes to Clearwater, and Princess Fenicia, thinking her husband had died, is shocked to see her husband (they are twins, she is mistaken). Fonzo goes the next day to rescue Canneloro, succeeds, and all ends well.
Fenicia, like Felicia, has no obvious nickname. Fen is pretty androgynous, Fenna very fern-like, Feni is cutesy, and Fee/Fi very short. However, it's possible for Fei/Fey to be a nickname.