What gentle memory did Orpheus bring back for Hades?

Orpheus attempted to bring Eurydice back to life by playing music with his lyre. Such music expressed his memory of Eurydice and his grief for losing her. This moved Hades and Persephone and they allowed him to take Eurydice back to life from the Underworld.

Explanation:

"Orpheus" was the son of Apollo and Calliope. He is known for his fateful love for his wife, Eurydice.

Although Hades allowed Orpheus to take Eurydice back to life, he was given a simple task of going ahead of Eurydice without looking back until he ascends into the world. When he was almost near the exit, he couldn't hear the footsteps of Eurydice anymore. He thought that he may have been fooled and so he looked back. This trapped the soul of Eurydice in the Underworld forever.