USA Today uses the phrase "auxiliary brain"

USA Today (!) has an article up about offloading long-term memory storage into services like Google: - This is the Google side of your brain
The article is intelligently written and while it does lean on the journalistic crutch of controversy ("A claims X, but B disagrees, saying ...") it remains remakably upbeat about the gradual shift of our notion of self to incorporate technology that's been going on for millions of years, ever since we first learned to build tools.