Possible origin suggested by John Ó'Néill - George and James Plunkett were muslin manufacturers on Carrickhill by 1819. In 1852 a George Plunkett had a pub at 118 Carrickhill beside the entry between Carrickhill & Union St known as Plunkett Court. So must be named after the family?
The Ulsterman - Monday 04 February 1856 carried a report from the Law Committee indicating it was agreed to attach spouts to some houses in certain streets, Plunkett Court being one of these streets.