I am from Dublin and have done some research into the Dockery name and where it originates.

In a book of Irish Names, I found it written as:(MAC)Dockery, where the Mac or Mc means 'son of'. The original form is Mac Dhail Re Dochair (Gaelic) and it came from east Connaught (Roscommon). It was corrupted to Mac Giolla Deachair (where deachair in Gaelic means 'hard'). In modern times, the Gaelic form is O'Dochraigh (where the O' also means 'Son of'). This as you can see is very far removed from the original form. The Mac Dockerys were one of the septs attached to the O'Connors in a military capacity.

If I find any more information, I shall post it in! Hope you found this interesting.