@CU123 MED was the initial HQ for the effort before much of the early research took place at Columbia. Columbia stands out as the organizing ground for the eventual involvement of several universities in many different locales. The project was moved from Manhattan because NYC was believed to be too vulnerable a location to conduct such confidential and high-risk research.
Two articles - both explain the importance of the project’s initial location in Manhattan and the contribution of Columbia towards the effort earning the name “The Manhattan Project”.
https://www.lanl.gov/museum/news/newsletter/2017/2017-06/manhattan-project.php