"Names can originate anywhere."

Quite true and also just because a name is 'for example' Anglian or Gaelic it doesn't mean the person who held that name came totally or even predominatly from Anglian or Gaelic stock or even from that stock at all. People got their names sometimes because of an association with a place, sometimes they were originally occupation names, and sometimes they simply assumed names. It is daft that we are even discussing it though. Whether someone's British ancestor from 1000 years ago had a Norman or Welsh name is hardly important as 1000 years of history within Britain makes one's family British!