Despite his desire to escape his father's realm, Prince Zagreus feels a deep connection to several of the characters he encounters throughout Hades. He can become friends with (and even romance) several, through deep conversation and gifts of nectar and ambrosia. To fully connect with the Underworld's denizens, however, the player must often complete a favor to grow closer to them or win their trust.
Some of these are part of the Fated List of Minor Prophecies, but others are more subtle, such as that for Hypnos. To form a bond with the God of Sleep, Zag must first facilitate some mediation between Hypnos and his brother, Thanatos.
Warning: Spoilers ahead regarding the full ending of Hades.
How To Initiate The Favor
In order to begin this diplomatic mission, the player must first have completed the game's main ending and persuaded Persephone to return to the underworld. Only during the epilogue can this story be completed.
Additionally, Zagreus must first increase his affinity with Hypnos by giving him at least 5 nectar. Players can acquire nectar as rewards for clearing chambers, or by purchasing it from the Wretched Broker for 10 Chthonic Keys.
Note that although Hypnos will repeatedly ask Zagreus for an autograph from the Minotaur (which he can eventually obtain for him), this does not count as Hypnos' favor and has no effect on his affinity gauge.
Talk To Hypnos & Thanatos
Once the above requirements are met, Hypnos will eventually reveal to Zagreus that a conversation with his brother Thanatos has made him feel as though the latter is displeased with his work. Zag, being close to the embodiment of Death, will agree to speak to Thanatos on Hypnos' behalf.
Once the two of them have this conversation, the player will occasionally come across the two sons of Nyx speaking to each other in the House of Hades. At this point, there's not much left for the player to do other than wait for the dialogue to exhaust and watch the scenario play out.
At first, Than is critical of Hypnos' work, and has little patience for his brother's lack of focus. Eventually, though, the two appear to come up with a method to help increase Hypnos' efficiency, and Thanatos ends up praising his improvement. Speak to Hypnos one last time, and he will thank Zagreus for his help. The final heart on Hypnos' affinity gauge will then unlock, allowing the player to gift him ambrosia and maximize their bond.