The Google logo as used from 1998 to 2015 was designed by Ruth Kedar. In an article from 2008, she comments on the design process, including the type choice:
Times Roman was the font of choice for the web at that time, while sans-serif fonts were the darling of the printed world. I wanted the readability of a serifed font, but looked for a typeface that had the same qualities we were looking for – subtly sophisticated, but with some humor and irreverence. The chosen typeface is based on Catull, an old style serif typeface. Catull borrows elements from traditional writing instruments such as the quill and the chisel with a modern twist. Search, by nature, is an activity that requires we look into the past. Therefore Catull’s historical ties seemed appropriate, as did the bridging between the old analog world and the new emerging digital era.
Shown below are design iterations created by Ruth Kedar, revealed in 2012. These were never publicly used. See also a slightly different evolution shown on Kedar Designs’ website in 2015.