This brought back some memories. In July 2004, I made a Firefox extension to "fix" a bad redesign of the website All Music Guide (my blog post: http://www.holovaty.com/writing/all-music-guide/). To my knowledge this was the first instance of a site-specific browser extension.

Aaron Boodman saw that idea and took it much farther by developing Greasemonkey, which made the idea more general and more powerful.