D&D 5e lore has precious little to say about the Lady of Pain. A search of DNDBeyond yields only 3 references in the entire 5th Edition corpus.
From the Dungeon Master's Guide:
The city is the domain of the inscrutable Lady of Pain, a being as old as gods and with purposes unknown to even the sages of her city. Is Sigil her prison? Is she the fallen creator of the multiverse? No one knows. Or if they do, they aren’t telling.
From Mordenkainen's Tome of Foes, in a section describing how the Raven Queen is thought to visit the souls of dead adventurers:
The Raven Queen’s reason for communing in this way is a matter of some dispute. Some sages posit that she is using people as pawns in an inscrutable game, the rules of which are known only to her and the Lady of Pain.
From a sidebar quote of a gnome living in Sigil, City of Doors, found in Mordenkainen's Tome of Foes, in the section on the Gnomish pantheon:
Who forged the chains that bind Tiamat in Avernus? Why do the modrons go on the Great March? Who is the Lady of Pain, really? I can’t tell you, but the answers lie in the Golden Hills. And if Garl and his gang don’t know, it can’t be known.
— Griballix, gnome of Sigil
This is every reference to the Lady of Pain in 5th Edition Material.
What resources from earlier editions of Dungeons & Dragons have more detailed exposition on the Lady of Pain?