The study of Near Eastern literature has the same value as the study of any other form of literature: it is the analysis of literary art for the sake of analyzing literary art. Though this may seem impractical to some, the fact of the matter is that art, good art at least, exists to shed light on the human condition. Therefore, ancient Near Eastern literature has as much value as one is willing to ascribe to the truths it is trying to present. If you can relate to the literature, then it is valuable.
The trouble with applying this to ancient Near Eastern literature, though, is that it was written by members of and for members of a society that is completely alien to the one in which we live. To try and relate to whatever point the author is trying to get across is an uphill battle from the start. However, at the same time, this may actually give the texts more value. For though we may not be able to relate to what is being said, we can understand that the most published texts must have been such because they were the most popular, and thus must have been what people at the time related to.
In this sense, Near Eastern literature has significant historical value. By understading what the average person in, say, Akkadian society valued, we are given a clearer picture of that society and what it stood for. This, in combination with other evidence such as architecture and archaelogical remains, can tell us a lot about a society that was part of the world and past that led to our world today. Though facts about Akkadian society in a vacuum may seem irrelevant, understanding the society as a whole and why it rose and fell can provide crucial information about our past and about how we can better function as a society. Like the old saying goes, "Those who do not learn from history are doomed to repeat it." By providing us with the values of this society, this literature is perhaps some of the most valuable evidence we have. And that is the why all good art is valuable; it provides small time capsules of how people, or even a person, at the time felt about the world around them or a specific subject. Without these, all we have are objective, physical objects that require us to do a lot of extrapolation that is, in most cases, unjustified. Art is history.