Here's my idea: the Demon Princes are sons to Overlords who rule over certain domains of the underworld:

-UR-JUBAAL, Demon Overlord of Madness. Reigns over the unbalanced, infuriated and unpredictable mind that is ruled by Chaos. A visionary and a fool, a prophet and a trickster.

-UR-TRAGGAL, Demon Overlord of Pain. Tormentor and victim, a channel for the constant searing agony suffered by both worlds. Enlightened and wise spirit.

-UR-VOMOCH, Demon Overlord of Voracity. Represents the insatiable appetite of Chaos; the uncontrollable need to move, to seek, to consume. Infinite greed or wanton lust are his domains, but also fiercely protective of that which is his.

-UR-AAZHEEL, Demon Overlord of Proliferation. The aspect of Chaos that represents uncontrolled growth and mutation; the force of creation but without system; order or structure. A kind father to his offspring.

-UR-KHRAG, Demon Overlord of Destruction. Representing the most obvious form of Chaos, he is the embodiment of one of its greatest powers: Entropy. The Demon Princes of Destruction generally lead the infernal legions to battle. They are honorable and strong warriors.

-UR-HEKAL, Demon Overlord of Hate. He is the patron of the wronged and the avenging ones, and has endless enmity against the universe and even his own creations. He reigns over impulsive fiery rage and cold, calculated hatred. The most “subtle” of the Demon Overlords, he governs the capital city which is named after him.