It is implied that the dragons Melisandre is talking about are part of the architecture of Dragonstone. As described by Salladhor Saan when Davos is pondering about the various dragon-shaped features of Dragonstone:
"If the red woman brings them to life, the castle will come crashing down, I am thinking. What kind of dragons are full of rooms and stairs and furniture? And windows. And chimneys. And privy shafts."
A Storm of Swords, Davos V
I do not think Stannis had any dragon eggs, because as Cersei says:
"If any such eggs remained, Stannis would have sold them to pay for his rebellion," the queen told him.
A Feast for Crows, Cersei VIII
Which would then imply that it was, in fact, stone dragons that Melisandre thought to wake. She most likely based this on the Azor Ahai prophecy that says that he shall wake dragons from stone:
"It is written in prophecy as well. When the red star bleeds and the darkness gathers, Azor Ahai shall be born again amidst smoke and salt to wake dragons out of stone."
A Storm of Swords, Davos III -- Melisandre
Since Melisandre is convinced Stannis is Azor Ahai, she assumes that's how the dragons will be awakened. However, we readers have already seen Daenerys being reborn in a fire, hatching her petrified dragon eggs.