It could be a habitational name from Ludwick Hall in Hertfordshire.
It could be the combination of a personal name (Luda) with a specialized farmstead (wic).
It could an anglicized variant of the Latin Ludovicus.
It could be the difference in speech clarity between "g" and "ck."
It could have come from the continent where Ludwig is a common surname.
I'm not sure where my study group of Lodwigs comes from. Maybe I will receive further enlightenment as I search further. They are fairly consistent with the spelling of Lodwig, but I have found in some census returns and other documentation a slip into Lodwick.
My starting person - Benjamin Lodwig Davies - came from Moylgrove, Pembrokeshire. There is at least 150 years of history here. I hope we can go further.