Harvard has been the most famous American college for the last two hundred years, according to Google’s nGram of books up through 2007.
From 1800 to 1820, the word Dartmouth appeared in books more often than the word Harvard. There was a foundational Supreme Court case in 1819 called Dartmouth College v. Woodward that was won by lawyer Daniel Webster.
Of course, one problem is that these references refer to more than just the college. I left out Columbia, which would be #1, because it comes up in District of Columbia, Columbia River, etc.
Berkeley was a popular word even before the U. of California, Berkeley was founded. The philosopher Bishop Berkeley was often written about as well as numerous other members of that aristocratic family in the British Isles.
Princeton is a town as well as a university, but it is striking how Princeton has outrun Yale since WWII.
Oxford and Cambridge dominate American colleges, with Oxford until the lead until the 1970s and Cambridge since then. Of course, Cambridge, MA is also home to two famous U.S. universities.
