As the title states, why does Melisandre believe that Stannis is Azor Ahai?
Because he wields a shining sword? Thoros of Myr swings a burning sword at tournaments...
The sword was even created by her own ceremony at Dragonstone and given to Stannis. This ceremony also provided the smoke and salt. A ceremony everyone could create by burning septs and plunging a sword in one of them, not exclusive to Stannis really. Did I miss something? I've looked at Is the Red Woman confusing Stannis for Daenerys? but the answers there doesn't really explain her original belief.
She creates all the signs of Azor Ahai herself and although she may be ambitious and can misread signs, something must have sparked the original belief that Stannis is him.
Answers from book canon with some logic speculation is accepted.