I looked this up and it seems that there was some talk about this ten years ago, but I'm not sure whether it ever got anywhere.
Civilization II had support for sub-year time designations, such as March 1942, and so on. If this feature does not currently exist in FreeCiv, it seems it would be nice to implement it. Going into making the calendars for each individual nation, etc., etc. is overkill, and scaling it down to allow single-day turns and accounting for leap years might be a little bit involved, but it doesn't seem like a simple twelve-month system should be too problematic. It'd also very be useful for creating historical scenarios where year turns make for very short conflicts (WWII, for instance, would be tricky to fully re-enact in five turns).