It refers more to the fact that Aegon V is the fifth Aegon to sit the Iron Throne/be the rightful ruler of Westeros. Therefore since Dany is claiming to be the rightful owner to the Iron Thron and ruler of Westeros, She can call herself Daenerys I, aka Daenerys, First of her name.
"The First of Her Name" signifies that Daenerys is the first queen named Daenerys to rule the Seven Kingdoms.
She is so called because she was born during a massive, notable storm that sank the Targaryen fleet off of Dragonstone. Literally, she was born in the midst of a storm, so she is called “Stormborn.”
Daenerys Targaryen, the First of Her Name, Queen of Meereen, Queen of the Andals and the Rhoynar and the First Men, Lord of the Seven Kingdoms, Protector of the Realm, Khaleesi of the Great Grass Sea, called Daenerys Stormborn, the Unburnt, Mother of Dragons.
Daenerys Stormborn of House Targaryen, the First of Her Name, Queen of the Andals and the First Men, Protector of the Seven Kingdoms, the Mother of Dragons, the Khaleesi of the Great Grass Sea, the Unburnt, the Breaker of Chains.
Drogon is the dragon that Daenerys herself rides. He's black and red and named after her late husband Khal Drogo.
'Stormborn' is simply an additional name given to her because she was born during a massive storm. It's not her last name. Matthew What Mitali said; Targaryen is her family and house name. "Stormborn" is an epithet or sobriquet of sorts, as she was born on Dragonstone during a gigantic storm.
Aegon I Targaryen, also known as the Conqueror, was the first Targaryen king of Westeros. He led his people to the island of Dragonstone to escape the Doom of Valyria.
The Targaryen family hails from Old Valyria, where lighter skin tones and hair colours were particularly common. As a result, the family had white or silvery blonde hair. The Targaryens have been able to keep their distinguishing Valyrian characteristics due to years of inbreeding.
During the wedding, exiled knight Ser Jorah Mormont pledges his loyalty to Daenerys while her benefactor Illyrio Mopatis gifts her three petrified dragon eggs. Daenerys is afraid of her new husband initially, but after learning the Dothraki language, she begins to bond with Drogo and genuinely falls in love with him.
TARGARYENS ARE NOT IMMUNE TO FIRE! The birth of Dany's dragons was unique, magical, wonderous, a miracle. She is called The Unburnt because she walked into the flames and lived. But her brother sure as hell wasn't immune to that molten gold.
Daenerys was born nine months after their flight, while a great storm raged above Dragonstone, sinking what remained of the Targaryen fleet; for this reason she is known as "Daenerys Stormborn".
Game of Thrones Daenerys Targaryen was pregnant by Jon Snow in finale ending | TV & Radio | Showbiz & TV | Express.co.uk.
Aegon I is the pinnacle of Targaryen greatness. His name is mentioned so often across the course of the show that book-averters and die-hard fans alike can both recite his history. He is the reason that the Seven Kingdoms exist as we know them. He is the legendary rider of the largest dragon, Balerion the Black Dread.
The phrase "blood of the dragon" refers to the Valyrian descent of the Targaryens, and more specifically to the fact that they were one of forty dragonlord families of Valyria.
Hailing originally from Old Valyria in Essos, the Targaryens are the only dragonlords to escape the Doom of Valyria, having moved their House to Dragonstone just a few years before the cataclysmic destruction of their homeland.
During the reign of King Viserys I Targaryen, it became truly customary to place a dragon's egg in the cradle of a newborn child. It was considered a sign that the child was a true Targaryen if the egg placed within his cradle hatched.
Why does Jon Snow have black hair? Because he isn't a “pure-blood” Targaryen: his mother, Lyanna Stark, has the same dark auburn, almost pure black hair as his, thus explaining, more or less, why his hair isn't Targaryen silver.
Most Targaryen's are born of incest. As most Targaryen's are blonde, that blonde gene remains prominent throughout most of the Targaryens. But Jon Snow's mother is Lyanna Stark who has dark hair, and brown/black hair will always be the dominant gene, so he was always likely to have dark hair.
Princess Rhaenys Targaryen, known as the Queen Who Never Was, was the daughter of Prince Aemon Targaryen and Lady Jocelyn Baratheon. She was married to Lord Corlys Velaryon of Driftmark.
The two surviving Targaryens, Viserys and Daenerys, fled into exile to the Free Cities of Essos across the Narrow Sea.
King Baelor the Blessed was the first Targaryen to suffer the Targaryen Madness. He was overzealously obsessed with religion and purity, to the point that he starved himself into an early grave because he believed that food is of this world, and the material world is sinful.
Aegon III, aka Aegon the Younger: Rhaenyra and Daemon's first child together.
King Aerys II Targaryen ("The Mad King")
Aerys II is a direct descendant of King Viserys I Targaryen through his daughter Rhaenyra.
Rhaenyra's three eldest sons all died fighting during the Dance of the Dragons, but her two youngest sons, the sons by her second husband, Prince Daemon, survived the civil war.