For clarification, what evidence would be needed to suppport the claimed date? I can only think of astronomy-related events for a pin-point date like Mursili's Eclipse (24 June 1312 BC)

Edit: Plugging in an earlier AskHistorian's thread similar to this question but with year. http://www.reddit.com/r/AskHistorians/comments/19pgf6/what_is_the_earliest_recorded_date_that_we_can/

Solar eclipses are useful because they're spectacular to see and would likely be recorded by anyone who could. A reasonably precise date was recorded in whatever system was in use at that time and place. We can then use mathematical models to find out exactly when and where a solar eclipse occurred that is the closest fit to the event described.

People in Ireland recorded a solar eclipse that we now know happened on November 30, 3340 BC.

source

This thread has some good answers:

What is the oldest specific date we know?