Why is Daenerys called Stormborn? If I recall correctly, she was born on Dragonstone rather than Storm's End (where Stannis Baratheon was rebelling).
On the night Daenerys was born on Dragonstone, a vast storm raged. For this reason she is sometimes called Daenerys Stormborn.
No squall could frighten Dany, though. Daenerys Stormborn, she was called, for she had come howling into the world on distant Dragonstone as the greatest storm in the memory of Westeros howled outside, a storm so fierce that it ripped gargoyles from the castle walls and smashed her father’s fleet to kindling.
-A Song of Ice and Fire: A Storm of Swords.
The night Daenerys was born a huge storm hit Dragonstone island. It was so powerful it sank what was left of the Targaryen fleet. Since then, Danny has been named Stormborn.