I don't know if I can comfortably talk about literary influence in the same way we talk about, say Milton's influence on the Romantic poets but we can certainly say that the Gilgamesh poems(that is, the Standard Babylonian and Old Babylonian poems and the Sumerian Gilgamesh "cycle") were very popular texts that were widely copied from their first writing down to the end of Cuneiform literacy and across a wide range of places. The text we often speak of, the Standard Babylonian epic, has been found in fragments at Nineveh, at Nimrud, at Assur, at Sultantepe in modern-day Turkey and in Southern Babylonia at Uruk and other major Southern cities; closely related texts which are somewhat earlier in date have been found at Emar in Syria and in Anatolia at Hattuša. It also seems to have been found in a wide range of contexts; while Assurbanipal's library was a royal archive of sorts the Sultantepe tablets come for the most part from a private archive and a closely related text from Nippur is part of a school exercise tablet. Moreover, the Standard Babylonian epic is only one "Gilgamesh". It is itself a substantial revision with large additions and re-arrangements (most famously the prologue) dating from the 13th-11th century BCE of an older Babylonian poem that we think based on tablets of it that survive was a good deal more compact and heroic in tenor. And these poems both inspired further variations; perhaps the most dramatic example of this is a prose retelling from Hattuša-someplace where cuneiform was very much a creature of administrative life and which was in some ways on the fringes of Mesopotamia's cultural orbit-in Hittite that revised the poem to focus on the episodes that took place in the Cedar forest, an area closer to and more familiar to the Hittites. We can see therefore that the story of Gilgamesh was at once canonical-something that people could and did learn in school-and something that authors and scribes could revisit, return to, and revise or retell in light of their own concerns. If you wish to research the history of Gilgamesh further, probably the best starting point is A.R. George's The Babylonian Gilgamesh Epic: Introduction, Critical Edition, and Cuneiform Texts or (more affordably) his paperback translation from Penguin Classics. Foster's The Epic of Gilgamesh is probably not as good a translation but has some good supplemental essays as well as other texts relating to Gilgamesh and showcasing the essay's lasting influence.