First Name
In the native Okinawan dialect, the yuna is a type of hibiscus flower. It is more precisely a sea hibiscus (hibiscus tiliaceus) which flourishes on all marine beaches of the world (but native to tropical Asia) and is employed in domestic medicine in the Caribbean. It's Yuna's main symbol: she wears a necklace with a pendant representing an hibiscus flower and the patterns on her skirt and obi look like this exact type of hibiscus flower, even if it's not a sea hibiscus.

As opposed to tidus which means "sun" in Okinawan, yuna means "night". Tetsuya Nomura has explained that even if all these kind of details may seem unnecessary, he does not want his designs to be meaningless. Obviously, sun and night are two opposites. Associated to Tidus, the sun may represent his cheerful character and personality. One can then deduce that associated to Yuna, the night may represent her cruel destiny and her strength of mind.

We can also take the symbol of the night and the sun as the Yin and the Yang. A woman and a man, one coming from one world, the other from another. Night relays sun as sun relays night. They need each other; the characters complement one another. Yin is said to embody feminity, mystery, night, unconscious, dream.

Saint Yuna of Brittany
Another curious origin unrelated to the name Yuna comes from Brittany, the northwest part of France. Her name day is August, 21st.

Chosen as a patron saint by clog makers, Yuna came from Wales with her brother Envel in the 6th century. There are many legends about the two saints but they are all related more or less to twinship. The most likely of them tells that they both set up an hermitage near Belle-Isle-en-Terre where they decided soon afterwards to never see each other again by act of contrition (will to turn away from sins and leave it to God). The story says that each day, Yuna would ring a bell at her prayers time. But one day, the bell stopped ringing and Envel understood that her sister passed away.

The name Yuna, deriving from Yves, has been gallicized by ecclesiastes and is given mostly yet hardly in Brittany today. Many different spellings follow from Yuna such as: Youna, Yeuna, Yoena, Jûna, Junan...

Last Name
In "Final Fantasy X", the characters don't have last names. Some characters seem to have one like Kimahri and Seymour but it's actually the name of the tribe they belong to. I don't really know why but maybe it has been done to create an emphasis on the "fantastic side" of the characters; to show that they don't belong to "reality" (i.e. in reality, we do all have a last name).

If Yuna had a last name, I would say it could probably be Braska, Yuna's father's name, because in Japan, people prevail last names over first names to talk to/about someone. It represents many things for them such as the family's fame and to which class they belong to.

Nicknames
During her pilgrimage, people respect Yuna a lot: they call her either Yuna sama (ユウナ様)/Lady Yuna or Shoukanshi sama (召喚士様)/Lady Summoner. In Japanese, those short particles such as sama (様), san (さん), chan (ちゃん), sempai (先輩) or kun (君) are used as suffixes after names to give a certain form of respect (depending on your interlocutor's age and sex); sama being the highest rank of respect a particle can attribute.

The nickname "Yunie" given by Rikku to Yuna in the English version doesn't exist in the Japanese version. It's just a fantasy from the English translators: they probably thought it would better fit Rikku's caring behaviour toward her cousin and reinforce their close relationship.

Pronunciation & Spelling
The only thing you must keep in mind is that the "u" of "Yuna" is a long vowel. Therefore, you must slightly extend the sound of the "u" when you pronounce it. Listen to the right and authentic pronunciation of Yuna, introducing herself to Tidus in the game: "ユウナです" (I am Yuna).

However, I wonder why they didn't spell her name ユーナ (Yu/—/na) instead of ユウナ (Yu/u/na) to lengthen the vowel "u" since it's how you regularly do it in katakana with the character "ー". Perhaps that's also why they should have romanized her name "Yuuna" instead of "Yuna", since the extra "u" seems not to be the lengthening of a vowel but an entire vowel by looking at the way they have chosen to spell it in Japanese.