Cradle of Echoes is a main quest in Horizon Forbidden West, which is started automatically (locking you out of all other quests and fast travel) after you deliver your first subordinate function to GAIA. So, that’s after you complete The Kulrut, The Sea of Sands, or Seeds of the Past — whichever you complete first.
During the mission, you can Varl investigate the Ninmah Research Lab in search of clues as to what’s been going on. Together you discover a large room full of ectogenic chambers, the like of which Aloy herself was made in. Proceed far enough north across this room and you’ll find a control console that you can use to access the storage units.
When you interact with the console, a disembodied voice will invite you to enter the container number you’d like to retrieve, and your objective switches to “Enter the Code”. You don’t have to find the Cradle of Echoes Code though, because you already have it. It was added to your Notebook automatically at the start of the quest. Go to your Notebook, then Datpoints, then Text Datapoints – Quests, and read the Distress Signal Datapoint (third row, fifth from the right). In addition to the coordinates of the facility, there is a three-digit number: 237. So, the Cradle of Echoes code in Horizon Forbidden West is 237. Enter this code into the console to find Beta, another clone of Elisabet Sobeck.