I have heard that the The Epic of Gilgamesh may predate the stories of Noah told in Abrahamic works. Is there any evidence to support which story came first?
The "standard" version of the Epic of Gilgamesh, the one edited by Sîn-lēqi-unninni, is dated to somewhere between 1300 BCE and 1000 BCE. Although the epic is far earlier, this is the version that includes Utanapishtim's story of the flood.
The dating of Genesis, however, as many matters that may have some religious significance is not entirely concluded. Depending on who you ask, you will receive answers ranging from 1400 BCE to the 5th century BCE, with the earlier limit of the range stemming from traditional views rather than scientific methods (e.g. the disputed concept of Mosaic authorship). If you accept the more conservative estimations of about 700 BCE, then the flood myth in Gilgamesh is clearly earlier.
If on the other hand, you wish to allow for the wildly exaggerated traditional datings, then I think you should also permit for a small switch of focus from Gilgamesh to the Akkadian Epic of Atra-Hasis. Atra-Hasis also includes a flood myth, and it's widely accepted1 that the flood myth in Tablet XI of the aforementioned "standard" version of Gilgamesh derives from it.
Furthermore, the flood myth in Gilgamesh is not directly comparable to the biblical narrative. In Gilgamesh, the tale of the flood is secondary to the eponymous hero's quest and is severely restricted by its first person narrative; Utanapishtim can only tell us what he personally experienced. One of the more important things we never learn, for example, is the reason for the flood, as, presumably, Utanapishtim was not aware of it.
Atra-Hasis, on the other hand, provides us with the history of the world before the creation of man, and a fuller story of the deluge, including an explanation of its cause. It is, therefore, a lot more comparable to Genesis than Gilgamesh, and this I believe is why we should focus on it instead. Following this path will lead us to an even clearer answer to your question as the earliest known record of Atra-Hasis can be dated to around 1646–1626 BCE (during the reign of Ammi-Saduqa), by its colophon. Thus, Atra-Hasis is earlier even if we accept Moses wrote Genesis.
Lastly, I would be remiss not to mention that the other Mesopotamian flood story, the Sumerian tale of Ziusudra, is also older than Genesis. The story is told in a single tablet, that has been dated to around 1600 BCE.
1 "In the intervening years, it is true, it had become clear that what Smith had discovered, the eleventh tablet of the Gilgamesh Epic, was derivative" - Moran, William L. “Atrahasis: The Babylonian Story of the Flood.” Biblica, vol. 52, no. 1, 1971, pp. 51–61. www.jstor.org/stable/42609705.
It's unknown, but it appears that although The Epic of Gilgamesh was written first, the Pentateuch (part of the Bible containing the creation stories) has the first description of the flood story.
From the Idaho University page on The Epic of Gilgamesh:
The oldest existing versions of this poem date to c 2000 BC, in Sumerian cuneiform. The more complete versions date to c. 700 BC, in the Akkadian language. The standard, first "complete" version, which includes the flood myth, is dated to c. 1300-1000 BC.
TL;DR: Oldest version of the poem is from around 2000 BC, the first version to depict the flood is around 1300-1000 BC.
In the book The City of God by Saint Augustine (best reference I can find), Moses is told to have been born in 1400 BC and died in 1201 BC.
Augustine also pens Moses as the author of the whole Pentateuch, so the flood story was written by him at some point during his lifetime.
So the Biblical flood story was written from around 1430-1201 BC, whereas the first version of The Epic of Gilgamesh to contain the flood story was written from 1300-1000 BC.
So it's possible either way, but it's most likely they were written somewhat concurrently.
Do note that it's unknown for sure whether Moses wrote the Pentateuch, but since the end of Genesis/beginning of Exodus (the first 2 books) discuss Moses, it's necessarily at this time period or after. (IIRC there is a 50 year gap between the books however)