IIRC (from memory) it was called Source because it was a derivative of the Quake 2 engine that id gave them a copy of after a meeting (on a cd simply labeled “source”) even before they had a contract in place.

If this apocryphal story is true, they might only have a license to it (making it a derivative work) and depending on the terms of their licensing agreement from id they might not be able to do that without having their legal people talk to ZeniMax’s (they bought out id software and are their parent company now) lawyers, which runs to thousands of dollars in pure costs even if everyone is totally cooperative and on board and wants to make it happen immediately.