Wyna Liu, the puzzle editor for Connections, brainstorms the categories first and makes sets of four words. Then she looks back over that and makes boards with some overlap, so it’s not just four distinct sets [of words]. We have an editing and testing process where other people read the boards and provide feedback.
In every board, there’s one category that’s a lot harder than the others: the purple category. Those words might have a more complex connection than the other sets. For instance, the purple category might be “Red ____” and the answers are things like “Red Herring” and “Red Sox.” So, at a first pass, “herring” and “sox” wouldn’t appear to have anything in common, but you have to make that harder, next-level connection.