My long-term-favourite prehistorical PoD might cause this... have the horses on the Eurasian steppe go overhunted and extinct instead of domesticated in the 5th/4th millennium BCE.
Proto-Elamite city states arose around the same time as Sumerian ones, so arguably before the domestication of horses could have affected Mesopotamia and Elam. And they were agriculture-based city states. Such polities would massively profit from the absence of nomads (or at least militarily strong nomads) from the mountains and across the mountains.
For later PoDs, I would put my money with more power and wealth with Elam`s traditional trading partners in the East, i.e. in the Indus Valley. In the late Neo-Assyrian Empire, there`s a reason why Aramaic came to replace Akkadian as the dominant language: the North-Western part of the Empire was a bridge to the socio-economically and politically increasingly important Mediterranean. Elam, on the other hand, increasingly became a backwater because it traditionally served as the bridge between Mesopotamia and the East, what we now perceive of as India. WIth "India"´s development taking place mostly elsewhere and the Indus Valley being a backwater itself over the last centuries of the 2nd millennium BCE and the early 1st millennium BCE, Elam was more or less on the fringe of the developing world, instead of in its centre. With a blossoming Indus Valley, Elamite may well have played the role which Aramaic IOTL played in the Neo-Assyrian Empire (if that empire ever would have come into being and crushed Elam in the first place).
That, in turn, would have greatly increased the chance of Elamite becoming the imperial language of a space-filling empire (the alt-Akhaimenids), just like Aramaic did IOTL. We still have Aramaic fringe groups IOTL today as a result. So maybe in this second scenario, even if nomads from the North (and later the South, i.e. Arabia, and then again from the North) build their various empires in the region, with Elamite being an important language in such a large area, there would still be speakers of Elamite around today, especially if they`re being identified with a religious minority group much later on (like Aramaic IOTL was with Christianity - of course the religious makeup of this world would be utterly different and unrecognisable).