What would be easy to implement and intuitive for the players would be for the game to display the date in relation to the season. That would mean "First/1st day of Summer", "Twenty-third/23rd day of Spring" and so on.
If you want to add it even more complexity you can also add the year, starting with the date in which the continent was founded, but I don't really see a need for that.
Edit: the advantage, btw, would be from a RP perspective. It would be natural to address the 3rd day of Autumn, but not so much the 11th of September.
