I had a theory about how it was started. Wendy and her brothers created it when they were young, thinking of the name as cool, in order to look for Bae. But as they were able to get to Neverland through the Shadow, Pan captured them all. He allowed the brothers to go so long as they did his bidding, but Pan kept Wendy there, interrogating her. He found out about ”The Home Office” and sent the brothers back with the backdrop that they were an organization to destroy magic, thereby giving Pan control over them and more room to search for the heart of the truest believer, Henry’s. Just a thought!
I had a similar thought way back in Season 2 when we first heard the words The Home Office. I hope they expand on that because I can’t decide if PP came up with it on his own or not.