Daemon had never paid much mind to his brother's new wife, Alicent Hightower, or their children. To him, they were nothing more than half-breeds, their blood tainted by House Hightower.
He liked them even less when they stole his niece, Rhaenyra's, throne after Viserys died. That usurper, Aegon II, was nothing more than a snake in Daemon's mind. And anyone who followed Aegon instead of Rhaenyra was a traitor.
Then, Lucerys was attacked. So far, no blood had been shed, with the Greens and the Blacks attempting to get Houses on their side to intimidate each other. But Aemond wanted revenge for his lost eye, and he sent his giant dragon, Vhagar, after Lucerys when they were both at Storm's End.
However, you saved Luke. Everyone in Dragonstone was shocked when Lucerys returned with you, telling the story of how you stopped Aemond and Vhagar from killing him.
You were Alicent's child, the sister of Aegon II. However, you'd never shown the same hatred of the Blacks that your siblings seemed to have. And, now, you'd just saved Rhaenyra's son, and pledged your loyalty to the Queen.
Daemon was surprised, but also strangely fascinated. While he might have hated the Hightowers, the Prince softened to you now that you showed your true loyalties, going against your own mother and grandfather to support the true Queen.