In Westerosi lore, the White Hart is a sign of royalty. That it appears to Rhaenyra and not the "celebrated" Prince Aegon highlights the show's exploration of fate versus political decree.
The episode centers on a massive royal hunt in the Kingswood to celebrate the second name-day of Prince Aegon, Viserys’s firstborn son with Alicent Hightower. House.of.the.Dragon.S01E03.PL.720p.WEB-DL.XviD....
In a desperate, near-suicidal solo charge to bait the Crabfeeder's forces out of the caves, Daemon secures a brutal victory. He emerges from the caves dragging the bifurcated torso of the Crabfeeder, cementing his status as a formidable (and rogue) power player. Key Character Shifts In Westerosi lore, the White Hart is a sign of royalty
Depicted as increasingly weary, struggling with the physical toll of the throne and the emotional burden of his choice of heir. In a desperate, near-suicidal solo charge to bait
Feeling replaced and pushed toward a marriage she doesn't want, Rhaenyra flees into the woods. Her encounter with a white hart —which she spares while the King's men kill a captured stag—symbolically validates her worthiness as heir. 2. The War for the Stepstones