Lithe was the name of the two days in the calendar of the Shire, that fell on either side of Midyear's Day. In the event of a Leap Year, the Lithedays were the day before Midyear's Day and the day after Overlithe (the extra day added in a Leap Year).

When translated to our modern calendar, Midyear's Day was on 22 June, so 1 Lithe was on 21 June and 2 Lithe was on 23 June. If it was a Leap Year, 2 Lithe would be on 24 June since Overlithe would then be on 23 June.
Encyclopedia entry originally written by Nienna-of-the-Valar