I saw a travel ad recently for Asheville, North Carolina that claimed that the New River that runs through town was the second oldest river in the world, second only to the Nile. I was surprised to hear this. I don't know why, I guess I was just surprised that something so old would be so close to where I live.
I decided to look into this. What I found was that there is no universally applicable and accepted method of determining the age of a river. About the best you can do is to put an upper and/or lower limit on the age. For instance, if you can determine through other methods when the rock that the river runs through first pushed its way up to the earth's surface, that would give you a maximum age. If you can date the sediment or fossils at the bottom of the river bed, that would give you a minimum age.
In many cases, these methods give you a very wide range that makes it impossible to accurately rank the ages of rivers. Here is a good article from the State of West Virginia that explains how difficult it is to determine the age of the New River (the New River runs through West Virginia and Virginia along with North Carolina). The info about the age is at the end, but the rest of the article is interesting as well.
According to the Journal of Geography, "Appalachia's New River was labeled 'the second oldest river in the world' during the 1970s as part of a campaign to save the river from being dammed. Despite the absence of geomorphic evidence, the promotion of the superlative age of the river was so successful that the mythical label became reality in the minds of the region's populace."