The term in reference to the haircut seems to have emerged into pop culture with the Beastie Boys song "Mullet Head."
Wikipedia also mentions that source:
According to the Oxford English Dictionary, the term mullet was "apparently coined, and certainly popularized, by U.S. hip-hop group the Beastie Boys", who used "mullet" and "mullet head" as epithets in their 1994 song "Mullet Head".
but also mentions the 1967 film Cool Hand Luke and a 1982 episode of TV show Cheers as sources.
The carlobarbershop entry for 'Mullet' says:
The origin of the term "mullet" has been traced back to the film "Cool Hand Luke." Apparently one of the guys calls people with long, shaggy hair "Mulletheads."
A set of articles on squidoo.com mentions some of the same sources and then comments:
Until that point, mullet-head was simply an old-fashioned American insult denoting a stupid or foolish person. Around since at least the mid-19th century, mullet-head may derive from the synonymous British slur mull-head, or from mullet, the name of a widely consumed, flat-headed fish. The term was even used by Mark Twain in his novel Huckleberry Finn.
A Guardian page called Why is a mullet hairstyle called a mullet? includes some relevant reader comments:
Other names are slightly more self-explanatory: the hairstyle is also known as the "Kentucky Waterfall" or the "Mississippi Top Hat" in parts of the US, and the "Bouncing Cobra" in parts of West Wales. In Germany it is known as the "Vokuhila" (vorne kurz, hinten lang= short at front, long at back). -- Anna, Cardiff
and the claim
The hairstyle was first worn by French fashion guru Henri Mollet in the early seventies. The "Mollet" did not see much light apart from in the french underground dance scene, until it was ressurected by popular television personalities such as Pat Sharp, the word having been anglicised by this point to "Mullet". -- Gary Badger, Scarborough, England
Mr Badger, Please do not confuse our non-British friends by attributing the words 'popular' and 'personality' to Pat Sharp. -- Joel Bradley, London England
Because it looks like you have a dead fish on your head? -- Janet Edwards, London, England
and a comment from "Mare" in Destin FL who claims to have created and popularized the term in 1987.