I see this stated in many places on the internet, but seldom referenced. The original claim was made in the book:
Lebot, Vincent. & Merlin, Mark David. & Lindstrom, Lamont Carl. 1992, Kava : the Pacific drug / Vincent Lebot, Mark Merlin, and Lamont Lindstrom Yale University Press New Haven
I don't have access to a copy of the book, but I was able to find this scholarly review by Timothy Johns. The reviewer seems to believe the claim of 3,000 years but noted some tendencies for the book to stray away from its data sources toward more of a narrative development.
"Kava, the root stock of Piper methysticum is, used to prepare a traditional psychoactive beverage of Melanesia and Polynesia, which is enjoying a renaissance in the Pacific and beyond. This volume examines kava, past and present, from multiple biological and cultural perspectives. It draws on data ranging from historical records to molecular biology and, as such, it represents a fine model of modern interdisciplinary and collaborative work in ethnobotany.
A consideration of the botany of kava follows a classical ethnobotanical approach in focusing on the origin of the domesticated species. Evidence from isozyme and numerical analyses complements information from taxonomy, morphology, geography, and linguistics to offer a convincing case that P. methysticum was domesticated in northern Vanuatu some 3,000 years ago from sterile clones of the wild Piper wichmannii and from there diffused throughout the Pacific Islands.
A chapter on the chemistry of kava combines a historical account with a thorough consideration of the physiology of kavalactones, the mildly narcotic and hypnotic active constituents of kava. Experimental data on the selection by humans for preferred kavalactone chemotypes during domestication provides a valuable complement to the previous chapter.
The chapter entitled "Ethnobotany: cultivation, classification, preparation, and medicinal use" furthers the development of the book as an integration of plant and human biology. This is unfortunately the least satisfactory chapter of the volume. I question its narrow view of ethnobotany as strictly indigenous knowledge as opposed to a field of scientific study; more unsettling is that in the process of describing aspects of cultivation techniques, folk classification and folk medicine, the chapter is uncharacteristically negligent in not providing descriptions of study methods or documentation of its sources of information.
Fortunately the scholarly form is regained in the subsequent chapter, which uses mythological and ethnographic data to offer insights into the religious and psychological attributes of kava that have made it a centerpiece of Pacific culture and society for generations.The following chapter examines the recent commercialization of kava. In the end the merging of the biological and cultural perspectives left me with a satisfying understanding of this intriguing case of human biology."
-TIMOTHY JOHNS, Center for Nutrition and the Environment of Indigenous Peoples,McGill University, Ste.-Anne-de-Belleviue, Quebec, Canada
Lebot's evidence for kava use back to 3,000 years ago is based on a synthesis of many different pieces of information. Arguments that combine numerical analysis, taxonomy, linguistics, geography, and morphology to draw a conclusion should probably be repeated with a bit more caution. After all, the book was clearly written for a wide audience, not scientists, and it's a book not a peer reviewed research article. For that reason, I'm not sure if we should blindly accept the statement that kava has been used for 3,000 years. That said, it's very likely that kava has been used for at least a thousand years or more in Vanuatu.