In October 680 the Prophet's grandson Husayn and about seventy companions were surrounded on the plain of Karbala, west of the river, and cut off from the water for three days before they were killed. That detail, rather than the battle itself, is what the Shia commemoration of Ashura turns on: thirst is the memory that organises the mourning, water is distributed publicly in Husayn's name during Muharram, and the free water stations found across the Shia world descend from it.
The lower Euphrates is also the homeland of the Mandaeans, whose religion requires living flowing water for its central rite of immersion, and whose communities lived for centuries along these rivers and marshes. Jorunn Jacobsen Buckley's work documents both their texts and their dispersal, which accelerated sharply after 2003; a religion built on a particular kind of running water is now largely practised far from it.
Christian tradition gave the river an end as well as a beginning. Revelation 16 has the sixth angel pour out his bowl on the great river Euphrates so that its water dries up to prepare the way for the kings from the east, an image that has been read into current events by end-times commentators for as long as there have been dams on it. The drying is now literal: upstream dams in Turkey and Syria have cut the flow substantially, and the southern marshes, drained deliberately in the 1990s and partly reflooded since, remain a fraction of their former extent.
Sources The Mourning of History and the History of Mourning: The Evolution of Ritual Commemoration of the Battle of KarbalaAli J. Hussain with The Mandaeans: Ancient Texts and Modern PeopleJorunn Jacobsen Buckley, The New Testament, RevelationTraditionally attributed to John of Patmos