A settlement of this name can be found on the maps of the National Library of Scotland, right back to the 1654 map by Blaeu. Indeed, Blaeu has a symbol that he normally used for a big(gish) house beside the name. It's not a church (they have a symbol and K) or a castle (he coloured them orange). It's not a plain farm - those I'm certain were farms back then all have an open circle. I'm wondering whether it's one of the peel towers that were all over the Borders at that time. There's certainly a lot of them in the area. I can only see one other settlement that MIGHT have a religious association to it's name (Ladycote), most of the names seem to be personal or geographical (eg Nisbeth, Boigend, Blacater).
There's nothing in the New Statistical Account about it.
I rather suspect that a guy called Mungo built his tower/fortified house there, and the name has stuck.