About this time legend among the Hobbits first becomes history with a reckoning of years.
What does "reckoning of years" mean in this passage from The Lord of the Rings?
It means they started to record their history and the passing of time.
The action or process of calculating or estimating something.
This fits with the rest of the sentence: "legend [...] first becomes history" when the telling of stories about past events and heroes becomes more concrete by fixing them in time, recording the passing of years and when the events took place, i.e. "reckoning [the] years".
To quote from another great work of epic fantasy: "Ages come and pass, leaving memories that become legend. Legend fades to myth ..." The difference between history and legend is often simply that legend is harder to pin down exactly but historical events have their own place in the calendar.