There are several answers to this question: The Wheel Stealer, Arnold, Sid (aka Sidney Monster) and Alistair Cookie.
In 1966, Jim Henson created three cookie eating monsters supposed to appear in a snack food commercial: The Wheel Stealer, the Flute Snatcher and the Crown Grabber. The commercial was made but never aired. Here it is on Youtube:
https://www.youtube.com/watch?v=5QngRWeEJGo
The Wheel Stealer was later seen as a munching monster in a Munchos commercial:
https://www.youtube.com/watch?v=xF7BlsSzHVQ
All three monster became recurring characters on Sesame Street: The Crown Grabber was later known as the Beautiful Day Monster, the Flute Snatcher turned into Snake Frackle. The Wheel Stealer, however, made his next career step around 1968 as Arnold, the Munching Monster in an IBM commercial
https://www.youtube.com/watch?v=7IgF6_jVaj8
before becoming the Cookie Monster on Sesame Street. Frank Oz (the monster's main puppeteer) later revealed that his name was Sid Monster.
http://news.bbc.co.uk/2/hi/entertainment/4432415.stm
By the way, on Sesame Street's official Twitter account, the monster calls himself Sidney.
https://twitter.com/sesamestreet/status/20796273130
Last but not least: When hosting the Monsterpiece Theater his name was Alistair Cookie:
https://www.youtube.com/watch?v=pudBqiTKA7U