As is always the case with me, I want parity with my biomes of the same general type. For the Desert, that means I want it to have just as much content as the Jungle, the current frontrunner for natural biome content.
In order to reach that level, the Underground desert still requires
A. A lair of some kind with a Non-hardmode boss. Similar to the Hive. An Antlion Pit with a giant Antlion boss gets my vote.
B. A Hardmode boss, like Plantera.
C. A Hardmode only mini-Dungeon with it's own boss. It seems like the Pyramid could very easily be converted into a Lizard Temple-sized mini-dungeon.