The identity of the world's first rheologist is most likely lost to the annals of time. Whether it was Ugg, Grugg, or their wives that were dragged by their hair into their caves after being clubbed over the head, we will never know who they were nor what they developed. A commonly cited candidate for the first rheologist is Heraclitus the Obscure, who stated "Everything flows" (Panta Rei), thus giving the Society of Rheology its motto.
But Heraclitus lived about 500 B.C., some thousand years after Amenemhet, who made a 7 degree change in the drain line of a water clock in order to compensate for the density differences in the water between day and night, and this was all done about 1600 B.C., although I'd argue that studying Newtonian fluid mechanics does not a rheologist make.
But there now appears a new claimant to the throne, an unidentified individual who back in 3500 BC developed a hair gel used by Egyptians. (Links for the research article, and a Nature News report (open access)). Being gels, they easily qualify as rheological materials, and hence there was a rheologist behind them, although who he/she was, we'll likely never know. (This also proves that human vanity has existed for ~ 5500 years).
All this raises the question of whether we'll ever seen an older example of rheology. Mummies don't get much older than this. Archaeological remains, however, do go back much further, but then trying to determine if a given artifact has rheological implications gets to be far more challenging. It is fun to think about though, isn't it?