By far the most lasting impact the Arameans had on the Middle East was the language that, via cultural diffusion, they imprinted on the ancient middle eastern societies. The Arameans inhabited the Fertile Crescent (the land between the Tigris and Euphrates rivers) in the 14th century BCE, but did not begin seriously influencing the region until three centuries later, when they began to spread into southern Anatolia and northern Arabia, which were Assyrian territories. The Arameans were a military force until about the 9th century BCE, when they fell to the attacking Assyrians. Although the Aramean nation fell, its language did not; Aramaic, which is very similar to Hebrew, was adopted not only by Babylonian Jews as the "Jewish tongue," but also by the well-informed as the language of choice. It was not until Greek emerged several centuries later that Aramaic lost its prestige as the most sophisticated language. Jewish practices still performed in Aramaic include the Ketubah (wedding contract), the Get (the divorce contract), and the Kaddish (mourner's prayer). Interestingly, much of Kabbalah (Jewish mysticism) is written in Aramaic. Also, the Talmud is written in a combination of Aramaic and Hebrew.
Sources: Bridger, David. Ed. The New Jewish Encyclopedia. NY: Behrman House, Inc. 1976.
Schreiber, Mordecai (ed.). The Shengold Jewish Encyclopedia. Shengold Books. 1998.
Telushkin, Joseph. Jewish Literacy. William Morrow and Company, Inc. 1991.