https://en.wikipedia.org/wiki/Harrington_(surname)
In Ireland, Harrington is an Anglicized form of the Gaelic name O’Harrachtain, meaning “the descendant of Arrachtan”, a nickname meaning powerful or mighty. It also could be an Anglicized form of O’HIongardail, later O’Hurdáil, meaning “descendant of longardal”. Lastly, it may be an Anglicized form of another Gaelic name: O’Hoireachtaigh, meaning “descendant of Oireachtach, a nickname meaning “member of the assembly”. The name is common in counties Kerry and Cork in Ireland.
It seems there is not an active clan organisation for your name, at this time.
However, the revival of interest in Irish culture and heritage over the past 100 years has led to the revival of many Irish Clans. In more recent times, increased access to historical archives online, DNA testing and ease of travel to Ireland has led many individuals to explore their own genealogy and connections to Irish Clans. Some of these have gone on to establish Clan Associations in Ireland, the USA, the UK, Canada, Australia, New Zealand and anywhere that the Irish Diaspora has found itself.
Clans of Ireland Ltd invite those interested in Irish Clans to contact them with a view to explore the possibility of reviving your Irish Clan.