In 1900, an unknown author wrote "The Goo-Goo Song".
In 1913, English writer Vincent Cartwright Vickers wrote "The Google Book", about the Google, a crocodile-like creature from Googleland.
In 1919, cartoonist Billy DeBeck introduced Barney Google, a comic strip character. His name supposedly came from the growing popularity of the word.
In 1923, Barney Google was written into a song, with "goo-goo-googly" emphasized. The word became popular with kids.
In 1940, Edward Kasner, a mathematician and professor at Columbia University, wanted to come up with a name for a large number, 1 followed by 100 zeroes, to make math fun for children (a noble, if futile idea). His nephew suggested "google", Kasner changed it to "googol", included it in his book, and it stuck. He also came up with "googolplex", or 1 followed by a googol of zeroes (now THAT's a large number).
In 1998, needing a name for their new company, Larry Page and Sergey Brin were looking for something that would signify a large amount of data. A friend suggested "googolplex", which was shortened to "googol", which was misspelled (potentially intentionally) as "google".